Letting $\bt = (\bt_1, \bt_2)$ partition the parameters, the restricted MLE (also known as the constrained MLE) is the value of $\bt$ that maximizes the likelihood under the restriction that $\bt_1=\bt_0$. In other words, some of the parameters are fixed at a hypothesized value $\bt_0$ while others are estimated.
In these notes, the restricted MLE is denoted $\bth_0 = (\bt_0, \bth_2(\bt_0)$.
Note that $\u_2(\bth_0) = \zero$, but $\u_1(\bth_0) \ne \zero$.
Theorem: If regularity conditions (A)-(D) hold and $\bt_0=\bts_1$ (i.e., if $H_0$ is true), then
\[\sqrt{n}(\bth_2(\bt_0) - \bts_2) \inD \Norm(\zero, \fI_{22}^{-1}).\]The proof is identical to the one for asymptotic normality of the MLE; only the notation has changed.
Note that this only works if $H_0$ is true: if it isn’t, there is no guarantee that $\bth_2(\bt_0)$ will converge to $\bts_2$.