Place 12 foil muffin cup liners in muffin cups. Place an amaretti cookie, rounded side up, into each liner.
Combine cream cheese, sour cream, and sugar in a large bowl; beat with a mixer at medium speed until light and fluffy.
Lightly spoon flour into a dry measuring cup; level with a knife. Add flour and extract to cream cheese mixture; beat at low speed until well blended. Spoon about 1/4 cup batter over each amaretti cookie. Bake at 325° for 27 minutes or until set. Cool completely on a wire rack (a slight indention will form).
Place peach pie filling in a small bowl; cut peaches into small pieces using kitchen shears or a knife. Spoon about 1 tablespoon peach filling onto each cheesecake.
Place remaining 3 cookies in a zip-top plastic bag. Seal bag; crush cookies with a rolling pin. Sprinkle cookies evenly over cheesecakes.