If making homemade pizza dough, prepare and rise as the recipe indicates.
Preheat oven at 500 degrees. Sprinkle cornmeal on a pizza stone.
While the dough rises, melt butter in a large skillet over medium low heat. Add the onions and rosemary, tossing with the butter and a big pinch of salt and pepper. Cook on low for 30 minutes, disturbing occasionally to prevent too-quick browning. If the onions begin to darken too quickly, lower the heat. In the final 10 minutes of cooking time, add the cider and cider vinegar. Heat on medium until mostly absorbed by the onions, which should appear jammy at this point.
Stretch, spread, or roll the pizza dough into the desired shape. Brush with 1 tablespoon of olive oil. Parbake the dough for 5-8 minutes.
To assemble the pizza: spread pesto onto the crust. Arrange the caramelized onions atop the pesto. Sprinkle on the Monterey Jack, havarti, and parmesan cheeses. Top with the thin pair slices and fresh rosemary, if desired. Bake for 15 minutes or until the cheese is bubbly, melted, and golden in parts.