Solution 1:

You are correct that Filet is the French spelling of fillet. According to the Oxford online dictionary:

Filet

French spelling of fillet, used especially in the names of French or French-sounding dishes

filet de boeuf

Fillet

a fleshy boneless piece of meat from near the loins or the ribs of an animal:

a chicken breast fillet

[mass noun] :

roast fillet of lamb (also fillet steak)

a beef steak cut from the lower part of a sirloin.

a boned side of a fish.

So, when preparing a dish, as an ingredient, you should use the word fillet. When referring to the name of the dish, it will depend (as you note: filet mignon).

Additionally, the Cambridge online dictionary states that filet is the US spelling of fillet:

Filet

US for fillet

Solution 2:

Norman Schur, British English A to Zed, third edition (2007) reports that fillet in British English has two corresponding senses in U.S. English: tenderloin and filet. Here is Schur's entry for fillet:

fillet, n. 1. tenderloin 2. filet

(Rhymes with MILLET, not MILLAY.) On an American restaurant menu the equivalent would be tenderloin steak, or perhaps filet mignon. The term may also be applied to pork, lamb, etc. 2. a piece of fish served without the bones.

Merriam-Webster's Eleventh Collegiate Dictionary (2003) treats filet as a variant spelling of one particular sense of fillet. Here is the Eleventh Collegiate's entry for fillet:

fillet in sense 2b also filet n {ME filet, fr. AF, dim. of fil thread, fr. L filum — more at FILE} (14c) 1 : a ribbon or narrow strip of material used esp. as a headband 2 a : a thin narrow strip of material b : a piece or slice of boneless meat or fish; esp : the tenderloin of beef 3 a : a concave junction formed where two surfaces meet (as at an angle) b : a strip that gives a rounded appearance to such a junction; also : a strip to reinforce the corner where two surfaces meet 4 : a narrow flat architectural member: a : a flat molding separating others b : the space between two flutings in a shaft

So in Schur's view, the main difference between fillet and filet is that, in addition to being coextensive with filet in a general sense in connection with cuts of meat, fillet in British English is used to refer to the specific cut of beef called, in U.S. English, tenderloin.

For its part, Merriam-Webster emphasizes that fillet and filet have the same meaning in meat sense, but then notes that fillet has multiple additional meanings that filet does not. On the other hand, according to the Eleventh Collegiate, filet has one meaning in English that fillet doesn't:

filet n {F, lit., net} (1838) : a lace with a square mesh and geometric designs

Aside from an entry for filet mignon (which Merriam-Webster defines as "a thick slice of beef cut from the narrow end of a beef tenderloin" and assigns a first occurrence date in English of 1835), that is the only entry for filet in the Eleventh Collegiate.

With regard to the word's etymology, Glynnis Chantrell, The Oxford Dictionary of Word Histories (2002) has this entry for fillet, confirming the summary that appears in the Eleventh Collegiate:

fillet {Middle English} Early use of the word was to denote a band worn round the head. Old French filet 'thread' based on Latin filum 'thread' is the source of the word in English. Current senses all involve the notion 'thin strip' (e.g., fillets of beef, fillet impressed on a book cover, etc.).

Solution 3:

In the USA I have most often seen "filet" for the noun (regardless of whether it is fish, beef, pork, etc.) although fillet is also "permissable". It is correct either way. The English culinary word derives from the French word like many or most culinary terms such as "purée". And, for example, the interchangeable word for beef tenderloin (filet mignon). When using it as a verb (i.e. to "debone" fish or meat) it is preferred to use fillet (present tense) and filleted past tense. Example: He filleted the red snapper with his fancy Japanese steel chef's knife. Ah. By the way, a knife specially created for deboning is sometime called a "filet knife", sometimes a "fillet knife". I see them both ways on Amazon.com. And to clarify. A steak (or piece of fish) with bone removed is a filet. One with the bone is more correctly called a chop. For example, a T-Bone and Porterhouse steak are not filets, they are technically chops. No offense intended to the butcher that stated the opposite for the noun and the verb. By the way, whether spelled filet or fillet, it is pronounced "filay".

Solution 4:

The difference in pronunciation is simple; spelling. If I see the word spelled filet on a menu I will pronounce it ‘filay’. If it is spelled fillet then it is pronounced ‘fillett’, hard t. That is because they are two separate words and should not be confused. If you wish to be pretentious by all means pronounce fillet as ‘filay’ but just know it’s wrong and people outside the US will laugh at you for it.