Remove paper carton from ice cream block. Place block on a long side on cutting board. Using sharp knife, slice off lengthwise six 1/2-inch-thick rectangles; return remaining ice cream to freezer. Lay slices flat on waxed- paper-lined baking sheet. Freeze 45 minutes, until firm.
Meanwhile, heat oven to 350°F. Stir cake mix, eggs, oil, cherries and extract in bowl until dough comes together. Divide into 24 pieces, about 1-1/2 tablespoons each. Roll into balls. Place 3 inches apart on ungreased baking sheets, about 8 balls per sheet. Flatten each into a 3-inch circle; if sticky, grease bottom of glass.
Bake in 350°F oven 10 minutes, until firm. Cool on sheet 2 minutes. Transfer cookies to rack to cool.
Remove ice cream slices from freezer. Using 3-inch round or scalloped cookie cutter, quickly press 2 rounds out of each ice cream slice, for total of 12 rounds. Save ice cream scraps for snacking. Using spatula, transfer 1 ice cream round onto 1 cookie; top with another cookie. Wrap in plastic wrap. Repeat, using all rounds and cookies. (If ice cream begins to melt, freeze.) Freeze.