Preheat oven to 450 degrees.
Using a small saucepan over medium heat, melt the butter and swirl it around a few times. It will foam and spatter. After about 3 to 4 minutes, it will start to smell nutty. Your brown butter is ready when the sizzling quiets down and you see little brown bits drop to the bottom of the pan.
Let cool for a bit and whisk in a vanilla bean and the vanilla extract. Set aside.
Roll out your pastry dough into about a 12-inch round. Transfer to pie plate and let the edges hang over. Cut off the excess pastry dough around the edge of the pie plate and crimp the edges all around. Alternatively, you can use a fork to press in all around the pastry dough edges as well. Prick the pastry dough with a fork, all over the bottom and edges. Add pie weights and bake the pastry crust for about 8 to 10 minutes, or until just slightly golden brown. Remove from the oven and let cool.
Once cooled, arrange the apple slices into the tart, making a circle with the apples and having them overlap. I used the granny smith apples for the outer edge but you can arrange however you like! Continue with more apples until the tart is full. Repeat with a second layer of apples on top. Brush the apples and crust with the brown butter vanilla mixture, discarding the vanilla bean. Combine the turbinado sugar with the cinnamon and sprinkle over the top.
Bake the tart until apples are golden brown and the filling is slightly bubbling, for about 25 to 30 minutes. Transfer to a rack and let cool. Serve warm and enjoy!