Make sauce: 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 goat cheese piece, 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 quesadilla, 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.
To Roast Peppers: Using a long-handled fork, char peppers over an open flame or on a rack set over an electric burner, turning them, until skins are blackened, 4 to 6 minutes. (Or broil peppers on a rack of a broiler about 2 inches from heat, turning them every 5 minutes, 15 to 20 minutes, or until skins are blistered and charred.) Transfer peppers to a bowl and let stand, covered, until cool enough to handle. Keeping peppers whole, peel them, starting at blossom end. Cut off pepper tops and discard seeds and ribs.
*available at specialty food shops and many supermarkets.
Recipe courtesy of Gourmet Magazine