Step 1: The target molecule 2-ethoxy-2-methylpropane is the ether \( (CH_3)_3C-O-C_2H_5 \), built from a tertiary tert-butyl fragment and an ethyl fragment joined through an oxygen atom.
Step 2: Williamson ether synthesis is an \( S_N2 \) reaction in which an alkoxide ion displaces a halide ion from an alkyl halide. The reaction works best when the carbon bearing the leaving group is primary, since a bulky nucleophile cannot easily approach a crowded tertiary carbon.
Step 3: If tert-butyl chloride is combined with ethoxide ion, the ethoxide would have to attack a tertiary carbon. This is sterically blocked for \( S_N2 \), so the strong, small base instead pulls off a proton and gives elimination (E2), forming isobutylene rather than the ether.
Step 4: If tert-butyl alkoxide is combined with ethyl chloride instead, the bulky tert-butoxide only needs to act as a nucleophile toward the unhindered primary carbon of ethyl chloride. This \( S_N2 \) substitution proceeds cleanly and gives the required ether.
Step 5: Therefore the correct route uses t-butyl alkoxide with ethyl chloride.
\[ \boxed{\text{t-butyl alkoxide + ethyl chloride}} \]