Find a differential equation for $w(t) = \phi_t(z) = \frac{z+ \tan t}{1 - z(\tan t)}$.
What about something like the following:
$$\begin{align} w(\tan(x)) &= \frac {\tan(x) + \tan(t)}{1 - \tan(x)\tan(t)} \\ &= \tan(x+t).\end{align}$$
Differentiating both sides with respect to $x$, using that $\tan'(x) = 1 + tan^2(x)$, this becomes
$$ w'(\tan(x))(1 + \tan^2(x)) = 1 + \tan^2(x+t),$$
i.e.,
$$ w'(z)(1 + z^2) = 1 + (w(z))^2.$$
Will this do? In fact, if there are no restrictions on what kind of a differential equation you need, you can probably just multiply the defining equation of $w(t)$ by the denominator and differentiate both sides, for example.