Is $T=\{\langle M\rangle \mid |L(M)| =1 \text{ or } |L(M)| >2\}$ recognizable?
You can establish a reduction $\overline{H}\le_M T$ by mapping $\langle M\rangle\rightarrow \langle N\rangle$ where

N(x) =
   if x = 0, accept
   if M halts on the empty string
      if x = 1, accept


Now 


If $\langle M \rangle\in H$ then $L(N) = \{0, 1\}$ which implies $\langle N\rangle\notin T$
If $\langle M \rangle\notin H$ then $L(N) = \{0\}$ which implies $\langle N\rangle\in T$


and since you already know $\overline{H}$ is not recognizable, $T$ must not be recognizable.