Preheat the oven to 300 degrees F (150 degrees C).
Melt butter in a 8x2-inch round springform pan in the preheated oven. Let cool 10 minutes.
Combine graham cracker crumbs and sugar in a bowl; mix well. Pat mixture into the bottom of the springform pan with butter.
Bake crust in the preheated oven until golden and set, about 10 minutes. Remove from the oven and let cool.
While crust is cooling, beat eggs using an electric mixer in a bowl for 3 minutes. Add 1 cup sugar and beat for 2 minutes more. Mix in cream cheese and cottage cheese slowly, adding one package of cream cheese at a time. Pour in 1 teaspoon vanilla extract; beat for 10 minutes. Pour filling into the cooled crust.
Bake in the preheated oven until firmly set, about 1 hour.
While cake is baking, mix together sour cream, 1/2 cup sugar, and 1/2 teaspoon vanilla extract for frosting. Set aside and allow to reach room temperature.
Remove cheesecake from the oven. Spread frosting carefully onto the top of the warm cheesecake.
Return cheesecake to the oven and bake until topping is golden and set, 10 to 12 minutes more.
Turn off the oven and allow cake to cool to near-room temperature in the oven, as this is key in preventing cracks or dips in cake top, about 30 minutes. Remove from the oven and chill in the refrigerator until ready to serve.