Is $E^{quasiP}$ equal $E$ or larger?
Suppose that $T(n) = n^{\omega(1)}$, and consider the complexity class $\mathsf{E}^{\mathsf{DTIME}(T(n))}$. Given a Turing machine $M$ and an input $x$ of length $n$, an $\mathsf{E}^{\mathsf{DTIME}(T(n))}$ machine can write $1^{2^n}$ and then ask the oracle whether $M$ halts on $x$ within $T(2^n)/\log T(2^n) = 2^{\omega(n)}$ steps. This cannot be done in $\mathsf{E}$ according to the time-hierarchy theorem.