Prepare the Baking Sheet: Line a medium-sized baking sheet with parchment paper and set it aside.
Make the Crust: In a mixing bowl, combine the flour, sugar, and salt. Attach the paddle attachment to your mixer and give the mixture a quick mix. Gradually add the cold butter cubes, mixing on low until the mixture resembles wet sand. Alternatively, you can use a pastry cutter to cut the butter into the flour mixture. Add the cold water one tablespoon at a time until the dough is firm and not sticky.
Prepare the Cheesecake Filling: Using a mixer or by hand, beat the softened cream cheese until smooth, about 20 seconds. Add the sugar and continue to mix until well combined. Add the vanilla extract and egg, and mix until just incorporated. Set the filling aside.
Prepare the Fruit Filling: In a bowl, combine the blackberries, blueberries, strawberries, sugar, cornstarch, and lemon juice. Toss until the ingredients are well incorporated.
Roll Out the Dough: On a flat surface dusted with flour, roll the dough into a circle approximately 12 inches (30 cm) in diameter. Transfer the rolled dough to the parchment-lined baking sheet.
Assemble the Galette: Spoon the cheesecake filling into the center of the dough, spreading it evenly but leaving about 3 inches (7.5 cm) of the edge free for folding. Gently pour the fruit mixture over the cheesecake filling. Fold the edges of the dough over the filling, creating pleats as you go to encase the filling while leaving the center exposed.
Finish the Crust: Brush the crust with egg wash and sprinkle with coarse sugar for added texture and sweetness.
Bake: Preheat your oven to 350°F (175°C). Bake the galette for 30-40 minutes, or until the center is mostly set and the crust is a golden brown.
Cool and Serve: Allow the galette to cool completely on the counter or in the refrigerator before serving. Top with whipped cream, powdered sugar, or a scoop of ice cream, and garnish with additional fresh fruit if desired.
Storage: Store any leftovers covered in the refrigerator for up to four days.