Preheat oven to 325 degrees. In a food processor or blender, combine cold butter, flour, brown sugar and salt. Pulse until it is incorporated and the pieces are tiny and course.
Press into the bottom of a 9x13 inch pan and bake for 12 minutes. Remove from oven.
To make the cheesecake layer:
Combine cream cheese, and sugar. Beat until smooth and creamy. Add the egg, milk, lemon juice, and vanilla. Pour cheesecake over warm crust and bake for 15 minutes. Remove and reduce oven to 300 degrees.
To make the Lemon Layer:
Whisk together eggs, sugar, lemon juice, lemon zest and flour until combined. Pour CAREFULLY over the warm cheesecake layer. If you pour too fast the cheesecake layer will lift. If it does lift a little, (mine did) it is ok! It will still taste amazing.
Bake for an additional 25-30 minutes until lemon layer is set. Remove from the oven and let cool. You can refrigerate for 3 hours and serve cold if desired. I think that way is best. Sprinkle powdered sugar on top and enjoy!