In a 9-inch cheesecake pan with removable rim (at least 1 3/4 in. tall), mix graham cracker crumbs and butter. Pat mixture evenly over bottom and about 1/2 inch up sides of pan.
Bake crust in a 350° regular or convection oven until slightly browner, 8 to 10 minutes.
In a 1- to 1 1/2-quart pan, mix gelatin with 2 tablespoons cold water; let stand until softened, about 1 minute. Stir over medium heat until gelatin is dissolved, about 45 seconds. Remove from heat and stir in liqueur and vanilla.
In a large bowl, with a mixer on medium speed, beat cheese, sugar, and gelatin mixture until smooth. Beat in eggs, one at a time, until smoothly blended. Pour mixture into hot or cool crust.
Bake cheesecake in a 325° regular or 300° convection oven until center jiggles only slightly when pan is gently shaken, 30 to 35 minutes. Cool cake on a rack for about 30 minutes, then chill, uncovered, until cold, at least 2 hours. If making up to 1 day ahead, cover cold cake airtight and chill.
Just before serving, lightly blot any moisture from surface of cake with a paper towel. Spread Persimmon Glaze over the cool or cold cake. Chill until firm, at least 15 minutes. To chill longer, cover cheesecake when glaze is set.
Persimmon Glaze: Rinse 2 or 3 soft-ripe Hachiya persimmons (each 2 1/2 to 3 in. wide, about 1 1/2 lb. total; see notes). Cut them in half and scoop out the soft flesh; discard stems and peels. In a blender or food processor, whirl flesh until puréed (you need 1 1/2 cups; save any extra for other uses). In a 1- to 2-quart pan, mix 1 tablespoon sugar with 1 1/2 teaspoons cornstarch. Add the 1 1/2 cups persimmon purée and stir over medium heat until mixture boils and thickens, about 4 minutes. Nest pan in a bowl of ice water and stir mixture until it's cool but still fluid, about 3 minutes. Immediately spread evenly over cool or cold cheesecake.