Place cream cheese and crack eggs in a large bowl and let sit at room temperature until the cream cheese is softened, about 1 hour. Arrange a rack in the middle of the oven and heat to 350°F. Line a baking sheet with parchment paper.

Cut ¼ inch off the top of the apples. Use a melon baller to hollow out the inside of the apples, leaving ½ inch of flesh around the bottom and edges.

Melt unsalted butter in a small bowl. Transfer 2 tablespoons to another bowl, add ¼ teaspoon of the ground cinnamon and ⅛ teaspoon of the ground nutmeg, and stir. Brush the inside of each apple with this mixture.

Add graham cracker crumbs to the remaining 2 tablespoons butter and stir until combined. Sprinkle a heaping teaspoon of this mixture into each apple.

Add granulated sugar, vanilla extract, remaining cinnamon, and nutmeg to the cream cheese and eggs, and beat until smooth. Spoon filling into each apple until filled to ½-inch from the top.

Bake until apples are softened and filling is golden brown, 30 to 35 minutes. Let cool completely before serving. Top with remaining graham cracker mixture and a drizzle of caramel sauce.