1. Preheat oven to 350°.
2. To prepare cheesecake batter, place cheese in a medium bowl; beat with a mixer at medium speed until smooth. Add granulated sugar and peppermint extract; beat well. Add 1 egg and 1 egg white; beat well. Add 1 tablespoon flour; beat mixture just until blended.
3. To prepare brownie batter, weigh or lightly spoon 4.5 ounces (about 1 cup) flour into a dry measuring cup; level with a knife. Combine 4.5 ounces flour, cocoa, and salt in a medium bowl, stirring with a whisk. Combine brown sugar, oil, buttermilk, vanilla, 2 egg whites, and 1 egg in a large bowl; beat with a mixer at medium-high speed until well blended. Add flour mixture to brown sugar mixture; beat at low speed just until blended.
4. Reserve 1/2 cup of brownie batter. Pour remaining batter into a 9-inch square baking pan coated with cooking spray. Carefully pour cheesecake batter over top; spread evenly to edges. Dot cheesecake batter with reserved brownie batter. Swirl top two layers of batters together using the tip of a knife. Bake at 350° for 26 minutes or until top is set. Cool completely in pan on a wire rack.