Combine flour, sugar, salt, and vanilla bean seeds in a food processor. Cut chilled butter into small cubes and pulse into the flour mixture about 10 times. Cut frozen butter into small cubes, add, and pulse just until a coarse meal with a few larger butter chunks forms.

Add ice water and pulse just until a dough forms, about 10 times. Turn dough out onto a countertop and form into a disk. Place between two large pieces of plastic wrap and roll out to a 1/2-inch thick round. Seal and chill at least 45 minutes or up to 2 days.

Unwrap the chilled dough and roll out on a floured surface to a 1/4-inch thick round. Cut into a 10-inch circle and discard scraps. Transfer to a lightly floured sheet of parchment on a baking sheet and chill for 30 minutes.

Heat oven to 350ºF. Toss sliced strawberries with sugar and cornstarch. Immediately arrange strawberries on the chilled dough in slightly overlapping concentric circles, leaving a 1-inch border. Carefully fold the dough edges up around the filling. Chill the assembled galette for 15 minutes.

Brush the dough with egg wash and sprinkle with 2 teaspoons sugar. Dot the berries with chilled butter pieces. Bake until golden brown and the filling is bubbly, about 45 minutes. Cool to room temperature on a wire rack before serving. Top with fresh whipped cream and snipped basil just before serving, if desired.