Heat oven to 350ºF. Put chicken breasts in a small baking dish and drizzle with oil. Season with salt and pepper. Bake until just cooked through, 10 to 12 minutes. When cool, cut into ½-inch-thick slices. Increase oven temperature to 375ºF. Lightly coat a large baking sheet with olive oil.
Dust a clean work surface with flour. Stretch pizza dough into an 11-inch-by-14-inch rectangle, with one long side parallel to work surface. Take breaks between stretches if dough starts to feel tight. Transfer to prepared baking sheet.
Scatter both cheeses over dough, leaving a 2-inch border at the edges. Scatter broccoli on top of cheese and press down on dough to adhere cheese and broccoli. Spoon marinara down center of filling.Sprinkle with red pepper flakes (if using). Place sliced chicken in a single layer down center of filling. Fold 1 short side of dough over filling, then fold the other short side over that. Fold top edge over center, then fold edge closest to you up and over to encase filling. Pinch along the seams to seal. Carefully turn over so stromboli is seam side down on baking sheet.
Beat egg with a splash of water. Brush stromboli with egg wash and sprinkle with sesame seeds. Using a sharp knife, create steam vents by slicing 3-inch slits across the top of the stromboli about every 2inches.
Bake until golden and bubbling, about 35minutes. Transfer to a cutting board. Let cool for 5 minutes before slicing.