Combine cream, basil, and sugar in a double boiler and stir until sugar dissolves, about 4 minutes.
Cover with plastic wrap, and refrigerate for at least 1 hour (or up to 2 hours for a more pronounced basil flavor).
Strain through a fine sieve into a bowl. Add mascarpone, and whisk until medium peaks form.
Cover, and refrigerate until ready to serve, up to 24 hours.
Make the fried basil: Heat oil in a deep skillet to 325 degrees.
Fry basil, a few leaves at a time, for 12 seconds. (The leaves will cause the hot oil to spatter- use splatterguard)
Drain leaves on paper towels, and let cool completely. Sprinkle both sides of the leaves with sugar. (Basil can be stored at room temperature overnight.).
Make the dough: Sift flour, salt, and sugar in a food processor to combine. Cut in butter, and pulse until mixture forms coarse crumbs.
Add ice water, and mix until just combined (dough will still be crumbly). Shape dough into a disk, wrap in plastic, and refrigerate for at least 1 hour (or overnight).
Preheat oven to 350 degrees.
On a floured surface, roll dough to 1/4 inch thickness. Cut out a 10-inch round, and transfer to a parchment-lined rimmed baking sheet. Refrigerate for 30 minutes.
Make the galette: Cut strawberries lengthwise into 1/4-inch-thick slices. Reserve end pieces for another use (want perfect even center slices for the concentric rings). Toss slices with 1/4 cup sugar and the cornstarch, and immediately arrange them in concentric circles on dough. Start 1 inch from edge, overlapping slices slightly. Fold edge of dough over berries.
Refrigerate for 15 minutes.
Whisk egg yolk and water o male eggwash. Brush dough with the egg wash, and sprinkle with remaining 1 tablespoon sugar.
Dot berries with butter.
Bake until crust is golden brown, 40 to 45 minutes.
Transfer to a serving plate. Serve warm with basil cream and fried basil.