Sodium ethoxide (C$_2$H$_5$ONa) and t-butyl chloride (C$_4$H$_9$Cl) can indeed be used to prepare t-butyl ethyl ether through a nucleophilic substitution reaction. Sodium ethoxide, being a strong nucleophile, can replace the chlorine atom in t-butyl chloride, leading to the formation of t-butyl ethyl ether. The reaction is as follows: \[ \text{C}_4\text{H}_9\text{Cl} + \text{C}_2\text{H}_5\text{ONa} \rightarrow \text{C}_4\text{H}_9\text{OC}_2\text{H}_5 + \text{NaCl} \] The appropriate starting materials for the synthesis of t-butyl ethyl ether are t-butyl chloride and sodium ethoxide.