Preheat your oven to 325°F (163°C). In a medium bowl, combine the graham cracker crumbs and 1/3 cup sugar. Pour in the melted butter and stir until the mixture is evenly moistened and resembles wet sand. Press this mixture firmly into the bottom of a 9x9-inch square baking pan, creating an even layer. Bake the crust for about 10 minutes until it’s lightly golden. Remove from the oven and let it cool slightly while you prepare the filling.
In a large bowl, beat the softened cream cheese with an electric mixer until smooth and creamy. Gradually add the 1 cup of sugar, continuing to mix until well combined and no lumps remain. Add the vanilla extract and mix again. Next, add the eggs one at a time, beating on low speed after each addition just until incorporated—overmixing can add too much air and cause cracks during baking.
Mix in the sour cream until fully combined, then sprinkle the flour over the mixture and gently fold it in. The flour will help the cheesecake bars hold their shape and prevent cracking.
Pour the cheesecake filling over the pre-baked crust and spread it out evenly with a spatula. Bake in the preheated oven for 40-45 minutes. The edges should be set, and the center will still have a slight jiggle when you gently shake the pan. This ensures the cheesecake remains creamy and not overbaked.
Remove the bars from the oven and allow them to cool to room temperature. Once cooled, cover the pan with plastic wrap or foil and chill in the refrigerator for at least 4 hours, preferably overnight. Chilling sets the cheesecake and makes slicing easier.
Use a sharp knife to cut the cheesecake into squares or rectangles. For clean cuts, wipe the knife with a warm, damp cloth between slices. Serve plain or top with fresh berries, fruit compote, or a drizzle of chocolate or caramel sauce for extra indulgence.