Is there any connection between 'biscuit' and 'bisque'?
Biscuit comes from the French word with the same spelling. The French word biscuit derives from bis meaning “twice” (from the Latin) and cuit meaning “cooked”, and it originally meant a type of bread that had been cooked twice and could be conserved for a long time, especially to serve as food during a long sea journey, but in modern French it most often means any kind of small, flat pastry such as a cookie, many of which are cooked a single time.
Bisque in the sense of a type of thick soup made with shellfish and cream also derives from the French word with the same spelling and meaning. The etymology of the French word is disputed. In the 17th century, it meant a type of soup with multiple different meats, often game and poultry, sometimes other variants such as carp, carp eggs and crayfish. Wikipédia cites Furetière's Dictionnaire Universel, which describes a bisque as a type of soup made only for very rich people, and gives the etymology as follows (my translation):
This meaning of the word comes from bis cocta, because since a bisque is made from several delicacies, it is necessary to make multiple separate and iterated cooking stages, until a last cooking stage that brings it to perfection.
The Trésor de la langue française cites a possible connection with the Normandy dialect word bisque meaning a type of low-quality alcohol and more generally a bad drink, and does not cite the bis cocta etymology at all, which casts some doubt regarding that hypothesis given that this dictionary is generally well-researched. The TLF goes on to cast doubts on a possible connection between the Norman word and Biscay¹.
Gilles Ménage's 1750 Dictionnaire étymologique (p. 199) has this to say (my translation):
Some believe that these soups were thus called because they were invented in Biscay. And as they are thick and viscous, having almost no broth, others have believed that they were thus names from viscus(https://en.wiktionary.org/wiki/viscus#Latin)². But honesty compels to say that the origin of this word is no better known than that of bisque in the context of the jeu de paume.
Ménage goes on to cite Le Duchat (I can't find the original work):
Might this meaning of the word bisque come from biscocta, short for biscocta offa? Or from bis sicca (also dropping offa)? What makes me suspect thus is that to make a bisque, one first pours a broth over the cuts [of meat], which is then let to simmer until the broth is consumed; then one pours once more a broth and let it simmer again; after which one serves the soup, which has thus become a bisque.
All in all, this makes me feel that the “cooked twice” etymology for bisque is possible, but by no means certain. And even if it is correct, it is an independent derivation from biscuit.
For bisque in the sense of a type of ceramics (which is also used in this sense in French, but isn't in any of the French dictionaries I could find), I can't find any good reference. I've found unsourced references to it being so named because it came from Biscay and others relating it to biscuit. Given that bisque is a specific type of ceramics (which, as far as I can tell is not fired twice) whereas biscuit refers generally to the result of the first firing, I think these words are unrelated.
For bisque in the sense of an advantage such as a free point conferred in a sport or game, the English word yet again derives from an identically spelled French word. In French, all sources agree that the word comes from the jeu de paume (a ball game, very popular in France from the late Middle-Ages to the early 19th century but now eclipsed by its variant tennis). The etymology of this meaning is also not known for certain, but at least the “twice cooked” hypothesis can be ruled out for lack of any meaningful connection. Ménage once again cites Le Duchat:
The origin of this word is as unknown as that of the Nile. (…) In la Maison des Jeux³ vol. 1, in the chapter on the jeu de paume, one reads in two places biscaye for bisque; which leads us to believe that this term came to us from Biscay, where perhaps the concept signified by this term was introduced before it was in France.
The Trésor de la langue française rates the origin as “obscure”, giving other citations for the biscaye form but refuting any relation with the Spanish province other than the identical spelling. The TLF suggests an etymological connection with biscaye meaning cheating at games or with the weight of sold products, which itself may be related to people from Biscay having a bad reputation or from the unrelated Italian word bisca meaning a gambling house.
¹ Biscay has little historical connection to Normandy, and a bisque is a pear alcohol which would be native to Normandy but not to Biscay. Note that in French, the Bay of Biscay is named after Gasgony, a different Basque region. Gasgony and Normandy were both part of the Plantagenêt kingdom, but Biscay wasn't.
² sic. Ménage cites thae Latin word meaning innards, not viscosus from which viscous was derived.
³ I don't know which of the several books with that name this refers to.
Originally, a bisque was thickened with rice and the ground shells of crustaceans. The word bisque, "crayfish soup" in French, stems either from the Bay of Biscay or the technique of bis cuites, or "twice cooked." Definitions of bisque - noun-- a thick cream soup made from shellfish https://www.vocabulary.com/dictionary/bisque
Gilles's answer is already well documented. I'd just like to add that the D.H.L.F.*, while mentioning the Biscay possible origin, favours another possibility, that given by Pierre Guiraud who establishes a relation between bisque and Provençal bisco: "small bevelled pieces" - related to Latin biaxius (with two slants)- the bisco being a soup containing small pieces of meat cut small (the orignal bisque was made with pieces of poultry, the "fish and/or crustacean" version appeared later). So according to that etymology the bis- in bisque would be related to the bis- in French biseau, bistouri or biais (cf English "bias").
*Dictionnaire Historique de La Langue Française