Preheat oven to 350°F (175°C) and grease a 9x5-inch loaf pan.
Beat the cream cheese and sugar together until smooth, about 2 minutes. Add the egg, vanilla, and flour, and beat just until combined. Chill in the fridge for 10 minutes.
Whisk together the flour, baking powder, baking soda, and salt in a bowl. Set aside.
Beat the softened butter and sugar until pale and fluffy, about 3 minutes. Add the eggs one at a time, then the vanilla.
Add the dry ingredients in three additions, alternating with the sour cream and milk, starting and ending with the dry ingredients. Mix just until combined.
Pour half the bread batter into the prepared pan. Spoon the chilled cheesecake layer evenly over the top. Dollop the remaining batter over the cheesecake layer.
Run a butter knife through both layers in a zigzag motion, 6–8 passes, to create a marbled swirl.
Stir together the graham cracker crumbs, sugar, and melted butter, and sprinkle evenly over the top.
Bake for 55–65 minutes, until a toothpick inserted into the bread portion comes out clean and the center springs back slowly when pressed.
Cool in the pan for 20 minutes, then transfer to a wire rack and cool completely, at least 2 hours, before slicing.