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Using a food processor, chop the garlic. Add the basil and with the machine on pour 1/2 cup olive oil and process until almost smooth.
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In a medium skillet, heat 1/4 cup olive oil over medium heat. Add the onion and cook, stirring occasionally, until softened and browned, 15 minute.
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Meanwhile, preheat a grill or grill pan to high. In a medium bowl, combine the beef and 1 teaspoons salt. Shape into 4 patties; brith with the remaining 1 tablespoons olive oil, then grill for 5 minute Flip and grill for another 5 minute.
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Spread the garlic mixture on the hamburger buns and top with a burger and some onions.