There's no way that your "solution for the PDE with another method" is correct.
Your equation can be rewritten as $$ u_y = \frac12 (x - u_x^2) $$ Evaluating on the set $\{x = 0\}$, you have $$ u_y(0,y) = - \frac12 u_x^2(0,y) \leq 0 $$
On the other hand, your initial data has $u_y(0,y) = -2y$, which is strictly positive when $y < 0$.
Your mistake is in the part you didn't show in the question.