Melt butter in a 10- to 12-inch frying pan over medium heat; add apples and sugar.
Cover and cook, stirring often, until apples just start to soften, about 10 minutes.
Stir in cornstarch mixture; cook, stirring gently, until thickened.
Remove from heat.
Beat together eggs, sour cream, and vanilla until smooth; combine with apple mixture.
Pour into crust-lined pan.
Bake in a 325° oven until center of cheesecake looks set and jiggles only slightly when pan is gently shaken, about 35 minutes.
Let cool, then cover and refrigerate until cold.