Preheat the oven to 350 degrees F (175 degrees C).

To make the crust: Beat butter and brown sugar together in a large bowl with an electric mixer until light and fluffy. Add flour and chopped nuts; stir until mixture becomes crumbly. Set aside 1/4 cup to use as a topping.

Press mixture into an 8-inch square pan. Bake in the preheated oven for 12 to 15 minutes. Let cool on a rack.

To make the filling: Beat cream cheese and white sugar together in a large bowl with an electric mixer until smooth. Stir in egg, milk, lemon juice, and vanilla until well combined. Spread filling mixture over baked crust. Sprinkle reserved mixture on as a topping.

Bake in the preheated oven until the edges are puffed and the surface is firm, about 25 to 30 minutes. Let cool on a wire rack before refrigerating until ready to serve.