Make the pastry cream: In a medium saucepan, warm 2 cups of the milk with half the sugar and the vanilla until steaming. In a bowl, whisk the egg yolks with the remaining sugar and the cornstarch until thick and pale. Stream the hot milk into the yolks while whisking, then return everything to the saucepan. Cook over medium heat, whisking constantly, until the whisk leaves a visible trail and the cream bubbles for 30 seconds. Stir in the butter, strain into a clean bowl, press plastic wrap onto the surface, and chill completely.
Blend the crepe batter: Put the flour, remaining milk (1 1/2 cups), eggs, 2 tablespoons sugar, salt, melted butter, and 1 teaspoon vanilla in a blender. Blend 20 seconds, scrape the sides, then blend 10 more. Cover and rest at room temperature for at least 1 hour.
Cook the crepes: Heat a nonstick skillet over medium and brush with butter or oil. Pour in a scant 1/4 cup of batter, swirl to coat the bottom thinly, and pour back any excess. Cook 60 to 90 seconds until the edges lift, flip, and cook 20 to 30 seconds more. Repeat to make 18 to 20 crepes, then cool completely before assembly.
Whip and fold: In a cold bowl, whip the heavy cream with the powdered sugar and remaining 1/2 teaspoon vanilla to soft peaks. Fold the whipped cream into the fully chilled pastry cream in three additions.
Prepare the strawberries: Hull and slice the berries about 1/8 inch thick. Toss with the remaining 1 tablespoon sugar and let sit 10 to 15 minutes. Pour off the released juice and reserve it separately.
Assemble: Set one crepe on the serving plate, brush with strawberry juice, spread about 2 tablespoons cream, and scatter 3 to 4 strawberry slices on top. Repeat, pressing down gently every few layers, until all crepes are used. Save the prettiest crepe for the top and tidy the edges with an offset spatula.
Chill: Cover the cake loosely with plastic wrap and refrigerate at least 4 hours, ideally overnight. The long chill is what makes the cake cut cleanly.
Finish and serve: Dust the top with powdered sugar and add fanned strawberry slices or drizzle the reserved juice over each slice. Cut with a serrated knife in one smooth glide, wiping the blade clean between cuts.