Prep pie crust.
Preheat oven to 375°.
Melt butter in a small saucepan over medium heat. Continue to heat; butter will foam and then subside. When butter begins to brown and smell nutty, remove it from heat. Transfer to a bowl to keep butter from continuing to brown. Set aside to cool.
Combine blackberries, brown sugar, salt, and vanilla extract in a large bowl. Pour cooled brown butter over berries. Mix well.
Place dough on a piece of parchment and top with another piece of parchment. Roll out dough to about a 14-inch circle or until dough is about 1/8-inch thick. Remove top parchment sheet and add berry mixture to the center of circle. Spread berries evenly over dough, leaving a 3-inch border.
Gently fold edges of dough in slightly on top of berries (pick up parchment edges to help assist folding the pastry over), creasing the pastry as needed. Carefully transfer parchment with galette onto a baking sheet.
Brush pastry crust with lightly beaten egg and sprinkle with turbinado sugar.
Bake at 375° for 20 minutes, or until crust is golden and fruit is bubbly. (if not using a rimmed baking sheet, you may want to put something to catch any pie drippings under the baking sheet.)
Allow to cool slightly. Serve warm with caramel sauce and vanilla ice cream.