Answer
2-bromo-2-methylbutane will react faster by the $S_N1$ mechanism.
Work Step by Step
We know that the structure which forms a more stable carbocation will react faster by the $S_N1$ mechanism. In the given pair, 2-bromo-2-methylbutane will form a $3^\circ$ carbocation, which is much more stable than the $1^\circ$ carbocation of ethyl iodide, even though ethyl iodide has a better leaving group. So, 2-bromo-2-methylbutane will react faster by the $S_N1$ mechanism.