Undecidability of TMs recognizing a decidable language
Given a machine $A$, construct a machine $M$ which on input $x$, acts as follows: run $A$ on the empty input, and if it ever halts, run the Turing machine encoded by $x$ on the empty input.

If $A$ halts on the empty input then the language $M$ recognizes is the halting language, while if $A$ doesn't halt on the empty input then the language $M$ recognizes is the empty language.