Light the grill and wait for it to heat up. Skewer a few pieces of chopped watermelon on a wooden skewer and stir together the melted butter and brown sugar. Brush the melted butter mixture over the skewered watermelon, transfer the watermelon and steak strips to the grill, and grill on both sides until the steak is cooked all the way through and the watermelon begins to brown slightly.
Unroll the sheet of puff pastry dough on a clean work surface and cut it into squares. Place the grilled steak, crumbled feta, and grilled watermelon in the center of 1 square of dough. Whisk 1 egg and brush the egg wash over 2 adjacent edges of the puff pastry dough.
Fold the uncovered edges of puff pastry dough over the filling to meet the edges covered in egg wash, and use a fork to press down along the edges to seal the dough. For the best results, transfer the puff pastry pocket to the freezer and freeze for 1 hour.
If the grill isn't already hot, light it again and wait for it to heat up. Grill the stuffed puff pastry pocket on indirect heat for about 8 minutes on each side.