For the strawberries: Put the balsamic vinegar and sugar in a heavy-bottomed saucepan. Bring the mixture to a boil over medium heat and cook until reduced by 2/3, about 30 minutes. Season with the pepper and some salt. Transfer to a small bowl and cool to room temperature, about 20 minutes.
For the puff pastry: Preheat the oven to 350 degrees F. Cut four hearts from the puff pastry with the cookie cutter. Put 2 of the hearts on a baking sheet. Divide the Brie slices between the hearts on the baking sheet, leaving a small border.
Mix the egg white with a splash of water and brush some around the edges of the filled hearts. Top them with the remaining 2 hearts and press the edges to seal. Brush the tops with egg wash, use a small paring knife to cut vents, and sprinkle with some coarse sea salt. Bake until puffy and golden, about 15 minutes.
To serve: When the puff pastry comes out of oven, toss the strawberries in the balsamic mixture and serve them on the side.