In a food processor or in a bowl with your fingers, whirl or rub flour and butter until mixture forms 1/4-inch lumps. Add cream and whirl or stir with a fork until dough holds together.
Pat dough into a ball, then divide in half. Flatten each portion into a 5-inch-wide disk. Wrap airtight and freeze until firm, about 15 minutes.
Sprinkle work surface with 6 tablespoons sugar. Dust rolling pin with 2 tablespoons sugar. Roll dough on sugar, a portion at a time, to 1/8 inch thick, turning over frequently and dusting with more sugar if needed to prevent sticking. Cut out cookies with a 1-inch-wide round cutter. Gather excess dough into a ball, reroll on more sugar, and cut out remaining cookies.
Place cookies 1/2 inch apart on buttered and floured or parchment-lined 12- by 15-inch baking sheets. Prick each round a few times with a fork.
Bake cookies in a 300° oven until golden, about 20 minutes; if baking more than one pan at a time, switch pan positions halfway through baking. With a wide spatula, immediately transfer to racks to cool completely.
Spread one side of half the cookies with vanilla buttercream filling. Top with remaining cookies.
Vanilla buttercream filling. In a bowl, with an electric mixer on medium speed, beat 3 tablespoons butter (at room temperature) with 1 1/4 cups powdered sugar and 1 teaspoon vanilla. Stir in enough milk, about 1 1/2 to 2 tablespoons, to make mixture smooth and thick enough to spread.
Nutritional analysis is per sandwich cookie.