In cleaving ethers with both SN1 and SN2 mechanisms possible, which occurs first?
One "side" of a molecule doesn't react by one mechanism and the other "side" by another mechanism. Generally speaking, a molecule reacts by a single mechanism.  Now, the mechanism can change depending upon the other reactants that are present as well as the reaction conditions.  In the case of t-butyl ethyl ether, if the reaction were carried out with $\ce{KI}$ under non-acidic conditions an $\mathrm{S_N2}$ reaction would occur and your answer (iodoethane and tert-butanol) would be correct.  On the other hand, if the reaction were carried out under acidic conditions then an $\mathrm{S_N1}$ reaction would occur.  As you pointed out a tertiary carbocation is more stable than a primary carbocation so the expected products in this case would be ethanol and 2-iodo-2-methylpropane.  Apparently, the reaction conditions in your reaction must have been such that the reaction occurred by the $\mathrm{S_N1}$ mechanism.