For the dough, place the first three ingredients in a food processor; cover and process until combined. Add the butter; process until mixture looks like cornmeal with bean-sized bits of butter, about 10 pulses. Add the egg and pulse 1 to 2 times.
Remove the bowl from food processer, remove the blade and bring dough together by hand. Place dough on lightly floured work surface and form into thick disk. Wrap tightly in plastic and refrigerate until firm, at least 1 hour.
Adjust an oven rack to the lowest position and heat oven to 400 F. Following refrigeration, roll the dough into a 12-inch circle about 1/8-inch-thick on a lightly floured work surface. Gently transfer dough onto a baking sheet lined with parchment paper or a baking stone.
In a separate bowl, mix together mango and strawberry slices, 2 tablespoons sugar, cornstarch and vanilla. Pour mixture in center of dough leaving a 2-inch border. Fold uncovered border over edge of fruit and pinch into pleats. Brush beaten egg onto crust and sprinkle with remaining sugar. Bake galette until golden brown and bubbly, about 30 minutes. Loosen with metal spatula and slide onto wire rack to cool slightly.
Place dulce de leche and orange liqueur in a saucepan and heat over low flame, stirring constantly, until hot. Remove from heat.
Serve galette warm with side of orange liquer-infused dulce de leche for guests to drizzle over their slice.