In a food processor, combine the graham crackers, granulated sugar, and melted butter. Pulse until the mixture resembles wet sand, about 30 seconds. Press this mixture firmly into the bottom of a 9-inch springform pan. Chill the crust in the refrigerator for at least 30 minutes.

Drain the canned peach slices and blend them until smooth. Reserve 1 cup of peach puree. In a large mixing bowl, beat the cream cheese and sugar together until creamy. Gently mix in the reserved peach puree, lemon juice, and vanilla extract, then fold in the whipped topping. Pour this filling over the chilled crust.

Cover the cheesecake and refrigerate for at least 4 to 6 hours, or overnight for best results.

In a saucepan, heat the remaining peach puree with a tablespoon of sugar until it simmers. Mix cornstarch with lemon juice until smooth, then stir into the peach puree. Cook for 2-3 minutes until thickened, then let cool before placing in the refrigerator.

Once set, release the springform pan and place cheesecake on a platter. Spoon the peach topping over it. Garnish with whipped cream or fresh peach slices if desired.