In a skillet cook onion and garlic in 1 tablespoon oil over moderate heat, stirring, until softened. Transfer onion mixture to a food processor and add roasted peppers, basil, remaining 2 tablespoons oil, and salt and pepper to taste. Blend sauce until smooth and transfer to a small bowl. Sauce may be made up to 3 days ahead and chilled, covered.
Put 1 tortilla on a work surface and spread with 1 piece of goat cheese, covering surface evenly. Spread 1 tablespoon pesto over goat cheese and cover with one fourth of onion. Repeat layering in the same manner, ending with a tortilla and gently pressing layers together. Make 1 more layered quesadilla with the remaining tortillas, goat cheese, pesto and onion. Spread top and bottom of each quesadilla with 1/2 tablespoon butter.
Heat a griddle or well-seasoned 10-inch cast iron skillet over moderately high heat until hot but not smoking. Cook quesadillas, 1 at a time, gently pressing down with a metal spatula, until golden, about 4 minutes on each side. Cut each quesadilla into 8 wedges and serve warm with roasted red pepper sauce.
*Available at specialty food shops and many supermarkets