Fast way to write out the utility optimization problem for a Cobb-Douglas function?
Well all Cobb-Douglas have an structure, it is the utility that makes shares constants as budget shares independent of prices 
$p_{l}x^{CD}_{l}(p,w)/w=\alpha_{l}$ with $u^{CD}(x)=\sum_{l}\alpha_{l}log(x_{l})$ and $\sum_{l}\alpha_{l}=1$. 
then you can go back and compute $x_l(p,w)$, with this the indirect utility, then invert to obtain the expenditure and then derive to get the hicksian. 
Not sure if this is what you want but some utility functions have a defining property, the same with the leontief or CES.