Preheat the oven to 350 ℉ (175 ℃) if not still preheated from the crust.
In a large bowl of a stand mixer, mix the cream cheese, corn starch, brown sugar and granulated white sugar together with a paddle attachment on medium speed for two minutes until everything is smooth and there are no lumps. (You can alternatively use a hand mixer). Scrape down the bottom and sides of the bowl to ensure everything has been mixed well.
32 oz (907 g) cream cheese, softened at room temperature, 3/4 cup (165 g) brown sugar, packed, 1/2 cup (100 g) granulated white sugar, 2 tbsp (16 g) cornstarch
Add in the reserved 1/2 cup (120 ml) of brown butter, vanilla, and sour cream to the cream cheese mixture and mix for 1 minute on medium speed. Scrape the sides and bottom of bowl again to ensure everything is mixed together.
1/2 cup (120 ml) brown butter from above, 1/3 cup (82 g) sour cream, at room temperature, 2 tsp vanilla
At this point, stop and put a pot of water on the stove to start boiling for the water bath. You will want about 1 ½ inches of water in the roasting pan.
Next add in the eggs and egg yolks to the cheesecake batter. Mix on medium-low speed just until combined. Do not over mix the batter.
3 eggs, at room temperature, 2 egg yolks, at room temperature
Pour the batter into the prepared pan with the crust.Place the cheesecake in a large roasting pan in the oven. Slowly pour the boiling water into the roasting pan around the cheesecake being careful not to splash the cheesecake.(If you're using the 10-inch pan method, place the cheesecake in the 10-inch pan, then place the 10-inch pan in the large roasting pan. Pour water into the roasting pan around the 10-inch pan.) Bake at 350 ℉ (175 ℃) for 1 hour. When 1 hour is up, turn off the oven and crack the oven door open using a wooden utensil. Let the cheesecake stay in the oven for 1 more hour. The cheesecake will continue to bake as it is cooling.
Once the 2 hours are up, remove the cheesecake from the oven and go around the edges of the pan with a butter knife to loosen the crust from the sides of the pan. Let the cheesecake cool completely on a wire rack. Cover with tin foil or plastic wrap, and refrigerate until cold, at least 8 hours.(I bake it the night before and let it refrigerate overnight).