In a large bowl, whisk together the eggs, milk, sugar, cornstarch, and gluten-free flour or potato starch until the batter is completely smooth. If any lumps remain, strain the batter through a fine mesh sieve.
Heat a wide nonstick skillet over medium heat and lightly grease with butter. Pour in a small amount of batter and immediately swirl the pan to create a thin, even layer. Cook for about 40 seconds on each side, then transfer the crepe to a plate. Repeat with the remaining batter. Refrigerate the finished crepes until fully cooled and easy to handle.
While the crepes chill, make the filling. In the bowl of a stand mixer fitted with the whisk attachment, combine the heavy cream, white cheese, cream cheese, sugar, vanilla pudding powder, and vanilla. Whip on medium speed until the mixture becomes thick and holds stiff peaks. Transfer the filling to a piping bag.
Pipe a strip of filling along the bottom third of each crepe. Fold in the sides and roll up into a tight blintz. Refrigerate until ready to serve.
Serve cold, topped with fresh raspberries or blueberries.