1. Infuse the basil flavor. Bring 1 1/2 cups cream to a gentle simmer in a small saucepan over medium heat. Bruise large basil leaves by hitting repeatedly with the dull side of a knife, and stir into cream. Remove from heat, cover tightly with plastic wrap, and let steep for 25 minutes. Strain through a fine sieve, pressing solids to extract liquid. Refrigerate until very cold, at least 5 hours.
2. Preheat oven to 400°F. In a bowl mix the dry ingredients. Whisk together flour, 3 tablespoons granulated sugar, the baking powder, and 3/4 teaspoon salt. Cut in butter with a pastry cutter or rub in with your fingers. In a separate bowl mix the wet ingredients. Whisk together 3/4 cup heavy cream, the eggs, and vanilla. Combine wet and dry ingredients using a fork until ingredients are moistened but not fully incorporated. (Do not overmix.)
3. Turn dough out onto a parchment-lined baking sheet, and pat into a 9 1/2-inch circle. Brush dough with remaining 2 tablespoons heavy cream, and sprinkle with 1 tablespoon granulated sugar. Bake until light golden brown and just cooked through, about 25 minutes. Let cool completely, about 2 hours.
3. Meanwhile, macerate strawberries with remaining 1/4 cup sugar, stirring occasionally, for at least 1 hour. Toss berries with small basil leaves. Whisk basil cream with confectioners' sugar in the chilled bowl of a mixer on high speed until soft peaks form. Mound basil cream on shortcake. Spoon berries with juice over top.