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.
Preheat oven to 400 degrees. 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. (The largest pieces should be the size of small peas.)
Whisk together 3/4 cup heavy cream, the eggs, and vanilla. Stir into flour using a fork until ingredients are moistened but not fully incorporated. (Do not overmix.)
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.
Meanwhile, macerate strawberries with remaining 1/4 cup sugar, plus more if needed, depending on sweetness, 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.
Shortcake is best served within 4 hours of baking but can be stored, untopped, at room temperature for up to 8 hours. Basil cream can be refrigerated for up to 1 day; whip just before using. Berries can macerate for up to 3 hours.