Is the torus the union of two connected, simply-connected open sets?

Edit: The answer is still negative. What you have to use is the Lusternik-Shnirelmann category $cat(X)$:

Definition. $cat(X)$ for a topological space $X$ is the least number of contractible open sets needed to cover $X$.

It is known that $cat(T^n)=n+1$, see here. Thus, you cannot cover 2-torus with two simply-connected open sets (since such sets are contractible).