Preheat oven to 350F and grease a 9-inch round cake pan lined with parchment. Beat gluten free flour, sugar, baking powder, salt, and softened butter together. Add eggs, milk, and vanilla extract and beat until smooth. Pour into the pan and bake 25 to 28 minutes until a toothpick comes out clean. Cool completely.
Make the pastry cream by heating milk in a saucepan until it just begins to simmer. In a bowl, whisk sugar, cornstarch, and egg yolks until pale. Slowly pour the hot milk into the egg mixture while whisking constantly.
Return the mixture to the saucepan and cook over medium heat, stirring constantly, until it thickens into a pudding-like consistency, about 3 to 4 minutes. Remove from heat, stir in butter and vanilla extract. Press plastic wrap directly onto the surface and refrigerate until completely chilled, at least 1 hour.
For the chocolate glaze, heat heavy cream until it just begins to simmer. Pour over chopped chocolate and butter in a bowl. Let sit 2 minutes, then stir until smooth and glossy.
Slice the cooled cake in half horizontally. Spread all of the chilled pastry cream onto the bottom half, then place the top layer back on. Pour the chocolate glaze over the top, letting it drip down the sides. Refrigerate 30 minutes before slicing.