\displaystyle \mathbb{E}[X|Y = y] = \sum_{x = 1}^6 x \, \mathbb{P}(X = x | Y = y) = \sum_{x = 1}^{y - 1} x \cdot \frac{2}{2 y - 1} + y \cdot \frac{1}{2 y - 1} = \ldots = \frac{y^2}{2 y - 1}