Place 1/2 cup butter into a large microwave-safe bowl; heat in a microwave until melted, about 15 seconds. Mix in crushed graham crackers and 2/3 cup white sugar. Press the mixture into the bottom of a pie plate, forming a crust.
Bake crust in the preheated oven until firm, about 10 minutes. Remove from oven; reduce oven heat to 350 degrees F (175 degrees C).
Combine brown sugar, 1/2 cup butter, milk, and vanilla extract in a saucepan over medium heat. Stir constantly until boiling; continue stirring until thickened, about 2 minutes more. Remove caramel sauce from heat and allow to cool.
Mix cream cheese and 1/2 cup white sugar together in a bowl until smooth. Pour in 3/4 of the caramel sauce and mix until well blended. Mix in all 3 eggs at once. Pour the batter into the crust.
Bake in the hot oven until edges darken but center is still a bit gooey, 35 to 45 minutes.
Turn off the oven, open the oven door, and let cheesecake cool in the oven for about 1 hour.
Reheat the remaining caramel sauce. Take cooled cheesecake out of the oven and pour warm sauce over. Let cool to room temperature, about 30 minutes. Cool 8 hours to overnight in the refrigerator.