Place graham crackers in a food processor and process into fine crumbs. Pour into a 12-inch springform pan. Add melted butter, 1/3 cup brown sugar, and 1 cup walnuts; mix until well blended. Press into bottom and up sides of the pan.
Combine 1 cup white sugar, 2 tablespoons water, lemon juice, and salt in a saucepan. Cook over medium heat, stirring constantly with a wooden spoon, until sugar is melted, about 5 minutes. Cook until golden brown, watching carefully to prevent burning, about 1 minute more. Remove from heat and carefully add heated warmed cream and 1 tablespoon butter. Pour caramel sauce into a measuring cup with a spout and allow to cool.
Pour 1/3 of the caramel sauce over the crust.
Bake crust in the preheated oven until set, 10 to 15 minutes. Remove from oven and set aside to cool; leave oven on.
Place diced apples in a bowl; add water to cover and stir in 2 teaspoons lemon juice to prevent browning. Drain well and pat dry. Combine apples with 1/3 cup brown sugar and cinnamon in a bowl; stir to coat.
Beat cream cheese in large mixing bowl using an electric mixer until fluffy. Add sour cream and 1/2 cup white sugar; beat on medium speed for 5 minutes. Add eggs 1 at a time, mixing well after each addition. Add flour and vanilla extract. Fold in cinnamon-coated apples and pour filling over crust.
Bake cheesecake in the hot oven until set, about 1 hour. Let cool in the pan for 30 minutes; release sides and place in the refrigerator until completely cool, about 30 minutes more.
Melt 1 tablespoon butter in a skillet over medium heat and add sliced apple. Cook until tender but not falling apart, about 2 minutes. Place slices around the edge of the cheesecake, creating a 2-inch ring. Sprinkle 1 cup walnuts in the center and drizzle remaining caramel sauce on top. Refrigerate until serving.