First, prepare the crust. Pre-heat the oven to 350 and liberally grease the bottom of a 9 1/2 inch springform pan with butter or shortening. In a bowl stir the graham cracker crumbs, melted butter and brown sugar together thoroughly until it resembles wet sand. Press the mixture firmly into the bottom of the pan until it makes a solid bottom layer. Bake the crust to let it set for 10 minutes. Remove it when it is done and set it aside to cool.
Next, prepare the filling. Reduce the oven heat to 325. Combine the cream cheese and sugar in the bowl of a stand mixer and beat them together until light and fluffy. Add the Greek yogurt and vanilla next. Once they are mixed in add each egg and egg yolk in one at a time. Finally, add the flour. Let the mixture get creamy and well combined.
Grease the inside sides of the springform pan liberally with butter or shortening. Pour the creamy cheesecake batter on top of the prepared graham cracker crust. Wrap the pan in foil to protect it. Then put the springform pan in an extra large roasting pan. Pour water into the roasting pan to surround the springform pan. Avoid getting water into the cheesecake and make sure the water comes to about a quarter of the way up the sides of the pan. This will prevent the cake from cracking.
Take the cheesecake out when it is done and let it cool for at least half an hour. Release it gently from the springform pan and carefully transfer it to a cake plate. Let it cool for an additional 30 minutes out of the pan. Cover the cake plate and refrigerate the cheesecake at least 4 hours to overnight. When ready to serve, spoon the strawberry balsamic sauce over the top and slice! Enjoy!