Whisk milk, sugar, cornstarch, and salt in a saucepan until smooth.
Cook over medium heat, whisking, until thick bubbles form, 3–5 minutes.
Remove from heat. Whisk in butter and vanilla. Spread custard in a shallow bowl.
Press plastic wrap on the surface. Chill until cool to the touch, about 30 minutes.
Heat oven to 400°F. Line a sheet pan with parchment. Lightly oil cone molds.
Unfold puff pastry on a lightly floured board. Trim edges. Cut into 1-inch strips.
Wrap each strip around a mold, overlapping slightly. Pinch seams to seal.
Brush pastry with milk. Roll or sprinkle generously with sugar to coat.
Arrange seam-side down on the pan. Bake until deeply golden, 18–22 minutes.
Cool 5 minutes. Gently twist molds out. Let shells cool completely.
Whip heavy cream and powdered sugar to soft peaks; keep chilled.
Beat mascarpone until smooth. Whisk it into the cooled custard until silky.
Fold in the whipped cream gently. Transfer filling to a piping bag with star tip.
Pipe cream into both ends of each shell. Serve soon, or chill briefly to set.