Melt 1 tablespoon butter in a small sauté pan. Cook onions until soft, about 5 minutes.
Lay 3 tortillas on a flat work surface. Spread each tortilla with 1/3 of the cheese, 1/3 of the Arugula Pesto, and sprinkle with 1/3 of the grated hard goat cheese and cooked onions. Top with remaining tortillas.
Warm a griddle or 10-inch cast iron skillet over medium-high heat. Melt a bit of remaining butter in the skillet and cook quesadillas, one at a time, until golden, about 4 minutes on each side.
Cut into 8 to 10 wedges and serve with Roasted Red Pepper Spread.