We give a simplified proof of regularizing effects for first-order Hamilton-Jacobi Equations of the form $u_t+H(x,t,Du)=0$ in $\R^N\times(0,+\infty)$ in the case where the idea is to first estimate $u_t$. As a consequence, we have a Lipschitz regularity in space and time for coercive Hamiltonians and, for hypo-elliptic Hamiltonians, we also have an H\"older regularizing effect in space following a result of L. C. Evans and M. R. James.