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1
In a skillet cook onion and garlic in 1 tablespoon oil over moderate heat, stirring, until softened.
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2
Transfer onion mixture to a food processor and add roasted peppers, basil, remaining 2 tablespoons oil, and salt and pepper to taste.
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3
Blend sauce until smooth and transfer to a small bowl.
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4
(Sauce may be made up to 3 days ahead and chilled, covered.)
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5
Put 1 tortilla on a work surface and spread with 1 goat cheese piece, covering surface evenly.
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6
Spread 1 tablespoon pesto over goat cheese and cover with one fourth of onion.
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7
Repeat layering in the same manner, ending with a tortilla and gently pressing layers together.
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8
Make 1 more layered quesadilla with the remaining tortillas, goat cheese, pesto, and onion.
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9
Spread top and bottom of each quesadilla with 1/2 tablespoon butter.
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10
Heat a griddle or well-seasoned 10-inch cast-iron skillet over moderately high heat until hot but not smoking.
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11
Cook quesadilla, 1 at a time, gently pressing down with a metal spatula, until golden, about 4 minutes on each side.
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12
Cut each quesadilla into 8 wedges and serve warm with roasted red pepper sauce.
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13
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.
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14
(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.)
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15
Transfer peppers to a bowl and let stand, covered, until cool enough to handle.
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16
Keeping peppers whole, peel them, starting at blossom end.
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17
Cut off pepper tops and discard seeds and ribs.