Heat the oven to 350°F. Stir the graham cracker crust ingredients together. Pour into a 9-by-13-inch baking dish and pack in very tightly. Use the flat bottom of a measuring cup and an offset spatula to flatten. (The tighter it’s packed, the less likely it will fall apart when cutting the cheesecake.) Bake until golden brown, about 15 minutes. Remove to a rack to cool completely.
In the bowl of a mixer fitted with the whisk attachment, combine the Neufchâtel, mascarpone, yogurt, powdered sugar, and vanilla. Whisk until smooth and fluffy, about 2 minutes.
Scoop the cream cheese mixture into the cooled graham cracker crust. Top with berries.
Cover with plastic wrap and refrigerate until set, at least 4 hours or preferably overnight.
To serve, place in the freezer for 10 minutes before slicing. Warm a utility knife or a metal bench scraper under hot running water. Slice and serve immediately.