Why utility rather than expected utility in Cochrane's "Asset Pricing"?
As mentioned in the comments this comes down to stylistic choices, since as you correctly pointed out:
$$u(c_t)+\beta E_t[u(c_{t+1})]=E_t[u(c_t)+\beta u(c_{t+1})]$$
However, in principle both expressions are correct. The first expression states that the $U_t(c_t, c_{t+1})$ is a composite function of present utility of consumption $u_t(c_t)$ and expected utility of future consumption $E_t(u_{t+1}(c_{t+1}))$, whereas the second expression is directly expected utility of present and future consumption. When we condition on presently known information those two are equivalent here.
The notation that Cochrane uses is not used everywhere  but there does not seem anything wrong with it either from mathematical perspective or economically.