To make the dough, in a large bowl, combine water, yeast and honey. Mix with a spoon, then let sit until foamy, about 10 minutes. Add in 2 1/2 cups flour and olive oil stirring with a spoon until the dough comes together but is still sticky. Using your hands, form the dough into a ball and work the additional 1/2 cup flour into the dough, kneading it on a floured surface for a few minutes. You can also do all the mixing and kneading in your stand mixer with the dough hook attachment. Rub the same bowl with olive oil then place the dough inside, turning to coat. Cover with a towel and place in a warm place to rise for about 1 1/2 hours.
About 30 minutes before you are ready to roll out your pizza dough start the onions. In a large skillet, melt the butter over medium-high heat. Add the onions and 1/4 teaspoon each salt and pepper, and cook, stirring constantly, until the onions begin to soften, about 5 minutes.
Stir in the brown sugar and cook, scraping the browned bits off the bottom of the pan frequently, until the onions are golden brown, about 20 minutes. Remove the onions and place on a plate, set aside.
To the same pan add a drizzle of olive oil and the broccoli. Cook about 10 minutes or until the broccoli softens slightly. Remove from the heat.
Add the shredded or chopped chicken to a medium size bowl. Add about half of the bbq sauce and toss the chicken with the sauce.
Preheat the oven to 450 degrees F.
Now grab the pizza dough and divide it into 4 balls (or 8 smaller balls for appetizer size). On a large floured work surface, roll out each pizza ball until you get about an 8-inch circle (or about a 6 inch circle for the appetizer size). Spread each circle with a little bit of BBQ sauce and then sprinkle with equal amounts of gouda and cheddar cheese. Top with equal amounts of the shredded BBQ chicken, broccoli, and then sprinkle with cilantro and parmesan. Tuck in the two sides of the dough and then roll up into a log, sealing the edges together by pinching the dough with your fingers. Place on a lightly greased baking sheet, and continue until all 4 (or 8) are complete.
Brush the tops of the strombolis with the egg wash. Now take a sharp knife and cut about 3 slits in the top of the dough to create steam vents.
Place the pan in the oven and bake for 10 minutes at 450, then reduce the heat to 375 and bake another 20 to 30 minutes or until the dough is cooked through and golden brown on top.
Serve with BBQ sauce for dipping.