To prepare cheesecake, combine gingersnap crumbs, butter, and milk in a medium bowl; stir with a fork. Firmly press mixture into bottom of a 9-inch springform pan coated with cooking spray. Bake at 400° for 3 minutes; cool on a wire rack. Reduce oven temperature to 325°.
Place cheeses in a large bowl; beat with a mixer at high speed until smooth. Add sugars, beating until well combined. Weigh or lightly spoon flour into a dry measuring cup; level with a knife. Add flour to cheese mixture; beat well. Add eggs, 1 at a time, beating well after each addition. Add sour cream, syrup, and vanilla; beat well. Pour cheese mixture into crust. Bake at 325° for 40 minutes or until cheesecake center barely moves when pan is touched. Turn oven off. Remove cheesecake; run a knife around outside edge. Return cheesecake to oven, and partially open oven door. Cool cheesecake 1 hour in oven. Remove from oven; cool completely on a wire rack. Cover and chill at least 8 hours.
To prepare compote, combine cranberries and remaining ingredients in a medium saucepan over medium-high heat; bring to a boil. Reduce heat, and simmer 5 minutes or until the cranberries pop, stirring frequently. Cool to room temperature, and serve over cheesecake.