"frozen minute"

You seem to have come upon some 19th century slang that hasn't survived. There are a few instances of the phrase that I was able to find and it's used in a variety of ways, not just one set meaning:

A Stiff-necked Generation By Lucy Bethia Walford

Seed of the Sun By Wallace Irwin

The Cornell Era Vol XXX edited by Henry Myers Bellinger et. al

Prison Screw by Leonard William Merrow Smith

The Saturday Evening Post, Volume 180, Issue 2

Most generally it seems to describe the feeling you get when time stands still because of nervousness or fear during an unsettling situation.

time stands still idiom

Cambridge dictionary

When time stands still, everything around you seems to stop:

I saw the car coming straight towards me, and for a moment time stood still.

In one of the other sources above, "frozen minute" seems to also describe the awkwardness of conversations where you don't know what to say, and it seems to last forever.

In the second to last source, "frozen minute" describes the time of suprise or petrification after a crash, allowing the criminals to capitalize on the victims' vulnerability.

In the last source, "frozen minute" is used in a completely different sense, to explain that just a minute on the dangerous rapids would be intolerable, because it would feel a lot longer due to the coldness of the water.

However, in the source in your description, "frozen minute" is used in the sense of "awkward conversation" that's found in the third source above: "...the frozen minute following an introduction? We seem to be experiencing its full iciness."