1Peel, quarter and core the apples. Cut each quarter into 3 lengthwise slices. In a large nonstick skillet, melt 3 tbsp. Butter over medium-high heat. Add the apples, 6 tbsp. Sugar and the lemon juice. Cook, turning occasionally, until the apples begin to caramelize, about 8 minutes. Lower the heat to low, add 1 tbsp. Butter and cook until the apples are tender, about 5 minutes. Let cool.
2Line 12 cups of a muffin pan with liners and preheat the oven to 325°. In a small pan, melt 2 tbsp. Butter, then place in a medium bowl. Mix in the graham crackers and salt. Divide the mixture among the muffin cups. Press into the bottoms with a narrow drinking glass. Transfer to the freezer to set.
3In a bowl, combine the cream cheese and 1/2 cup sugar. Using a mixer, beat at high speed until creamy, about 2 minutes. Beat in the eggs 1 at a time, scraping down the bowl twice, until blended. Mix in the vanilla.
4Arrange 3 apple slices in a single layer on the crust in each muffin cup. Reserve the remaining apples with their juices at room temperature in a covered bowl. Divide the cream cheese mixture evenly among the cups; bake until set but slightly jiggly, about 15 minutes. Turn off the oven, open the door slightly and let cool for 30 minutes. Refrigerate for at least 5 hours or overnight.
5In a small skillet, melt the remaining 2 tbsp. Butter over medium heat. Add the walnuts and 2 tbsp. Sugar and cook, stirring, until toasted, 4 to 5 minutes. Cool on paper towels, then break up the candied nuts.
6Arrange the reserved apples with their juices on top of the cakes. Stir the remaining 1 tbsp. Sugar into the sour cream. Top the cakes with the sour cream and candied nuts.