For the Browned Butter Crepes:
For crepe batter, melt the 2 tablespoons butter in a small saucepan over medium heat until light bronw. Immediately remove from heat. Skim off solids; set aside.
Combine the eggs, milk, and 1/2 cup water in a blender and blend on medium speed until smooth. Add the flour, cornstarch, and salt. Add the browned butter and vanilla extract. Blend until smooth. Refrigerate the batter for 30 minutes.
Melt a pat of butter in a 6-inch nonstick skillet over medium heat. Pour a scant 1/4 cup batter into the skillet, tilting the skillet so the batter covers the bottom in a thin layer. Cook until the crepe is lightly browned, about 1 1/2 minutes. Loosen a corner of the crepe, turn over, and cook until very lightly browned, another 15 seconds. Transfer to a plate. Continue cooking crepes, stacking until there are at least 12, adding more butter to the pan as necessary. Cool to room temperature.
For the Lemon-Berry sauce:
Melt the butter in a medium saucepan over medium-low heat. Remove from the heat and whisk in the sugar, lemon zest, and lemon juice. Add the egg and whisk until smooth.
Return the saucepan to medium-low heat and cook, whisking constantly, until the sauce thickens and coats the back of a spoon, about 5 minutes. (Do not boil or the eggs will scramble.) Remove from the heat. Whisk in the cream and set aside to cool.
Beat the cream cheese and yogurt in a medium mixing bowl with an electric mixer on medium-high speed until combined.
To serve, spread about 2 ½ tablespoons of the filling on one half of each crepe. Fold crepe in half over the filling, then fold almost in half again. Place two filled crepes on each serving plate.
Top each serving with a generous tablespoon of lemon sauce and some fresh berries. Serve immediately.