Why is "ten-minute break" singular but "two weeks' time" uses a plural?

We say a ten-minute break
We say two weeks' time.

Ten-minute is singular but two weeks' is plural, why?

It all depends on the countability or uncountability of the noun. "time", as used in your phrase, is uncountable, while "break" is countable (notice you say "a ten-minute break).

With words used in such phrases which can be either countable or uncountable, we can find both variants:

a. Use of the genitive (the noun has to be uncountable):

• I had three weeks' vacation.

b. Use of a compound noun formed by numeral+singular measure of time (with countable nouns):

• I had a three-week vacation.

In your examples, "time" is uncountable and thus uses pattern (a), while "break" is countable and uses pattern (b).