To prepare the cake, coat a 10-inch tube pan with cooking spray; dust with 2 tablespoons flour.
Lightly spoon remaining 3 cups flour into dry measuring cups; level with a knife. Combine 3 cups flour, baking powder, baking soda, and salt in a bowl, stirring well with a whisk. Combine 2 cups granulated sugar and butter in a large bowl; beat with a mixer at medium speed until light and fluffy. Add eggs, 1 at a time, beating well after each addition. Add flour mixture to sugar mixture alternately with buttermilk, beating at low speed, beginning and ending with the flour mixture. Add rind, juice, and mint; beat just until blended.
Spoon batter into prepared pan; sharply tap the pan once on counter to remove air bubbles. Bake at 350° for 45 minutes or until a wooden pick inserted in the center comes out clean. Cool in pan 10 minutes on a wire rack; remove from pan. Cool completely on a wire rack. Sift powdered sugar over top of cake. Cut cake into 16 slices.
To prepare compote, combine cherries, 1/4 cup granulated sugar, 2 tablespoons water, and cornstarch in a medium saucepan; bring to a boil. Cook 1 minute, stirring constantly. Remove from heat; stir in extract. Cool. Serve with cake.