 Solution

Let the stored bits be $G,H$, with $G$ most significant. The required Gray code transitions giveConnect $H$ directly to the first D flip-flop's data input and feed $G$ through one NOT gate to the second input. Drive both enable inputs with $E$ and both clock inputs with the same clock. Thus $E=0$ holds both stored bits; $E=1$ takes the next Gray-code counter state at the active clock edge. Reset initializes $GH=00$.

With ordinary uncomplemented $Q$ outputs, one NOT gate is minimal: wiring alone cannot produce $\overline{G}$. If a D flip-flop already provides a complementary output, that output supplies the inversion without an external logic gate. This is a Moore machine because its output $GH$ depends only on stored state; $E$ controls transitions.