Pulse the flour, sugar and salt in a food processor until combined. Add the butter and pulse until it resembles coarse meal mixed with pea-size bits of butter, 10 to 15 times. Add the egg mixture and pulse until the dough just starts to come together, about 10 times. Turn the dough out onto a piece of plastic wrap, using the plastic to help form it into a disk. Wrap tightly and refrigerate until chilled, at least 1 hour or up to overnight.
Whisk together the cream cheese, 2 tablespoons sugar, vanilla, egg yolk and salt in a medium bowl until smooth. Cover with plastic wrap and chill in the refrigerator, about 45 minutes. Toss the strawberries, cornstarch and remaining 1/3 cup sugar together in a medium bowl and set aside.
Position a rack in the center of the oven and place a rimmed baking sheet on it. Preheat the oven to 375 degrees F.
Sprinkle the dough lightly with flour on both sides and roll out on a lightly floured surface into a 12-inch round that is about 1/8-inch thick. Transfer the dough to a piece of parchment and place on an inverted baking sheet.
Spread about 1 cup of the cream cheese mixture over the dough, leaving a 2-inch border on all sides. Place the strawberry mixture in the center of the cream cheese and spread into an even layer over the filling. Dollop the remaining cream cheese mixture over the strawberries. Fold the edges of the dough over the filling, pleating as needed. Brush the crust with the egg white and sprinkle with coarse sugar. Refrigerate for 15 minutes.
Slide the galette and parchment paper onto the preheated baking sheet and bake until the crust is golden and the filling is bubbling, about 40 minutes. Transfer to a wire rack to cool slightly before serving, about 30 minutes.
When measuring flour, we spoon it into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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