Frozen Peppermint Cheesecake with a Brownie Crust

package (18.4 ounces) chocolate fudge brownie mix

large eggs

vegetable oil

water

package (8 ounces) cream cheese (at room temperature)

sweetened condensed milk

and 1/2 to 2 teaspoons peppermint extract (do not use mint extract)

candy canes (crushed)

Red food coloring (optional for richer color)

container (8 ounces) frozen whipped topping (thawed)

Whipped topping (optional)