Warm 4 tablespoons olive oil in a small saucepan over medium-low heat. Add the minced garlic and a pinch of crushed red pepper flakes and cook, stirring occasionally, until the garlic is just golden. Turn off the heat and reserve the garlic oil.
Season the chicken breasts all over with kosher salt and pepper. Brush a little of the reserved garlic oil on the chicken.
Heat a grill pan over medium-high heat. Grill the chicken 5 to 6 minutes per side until cooked through and an instant-read thermometer registers 165°F (74°C). Remove and let rest 10 minutes.
While the chicken rests, preheat the oven to 425°F (220°C) and line a baking sheet with parchment paper.
Slice or chop the rested chicken into bite-sized pieces.
Toss the broccoli florets with 1 to 2 teaspoons of the garlic oil and a pinch of salt and pepper.
Place the puff pastry sheet on the prepared parchment. Prick the pastry all over with a fork to prevent large bubbles.
Spoon a few tablespoons of the garlic oil mixture over the pastry, spreading evenly and leaving a 1-inch border for the crust. Sprinkle 1 cup of the shredded cheese over the pastry.
Top with the chopped grilled chicken and seasoned broccoli. Sprinkle the remaining 1/2 cup shredded cheese over the top and brush the pastry edges with the beaten egg wash.
Bake for 18 to 22 minutes, until the pastry is golden, puffed, and flaky. Remove from the oven and sprinkle with Parmesan cheese.
Let the pizza cool slightly on the baking sheet, then slice and serve.
