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Using your blender or immersion blender, add garlic, egg yolks, lemon juice, and vinegar. Blend until smooth.
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In a very slow stream add your olive oil while blending simultaneously. Blend until smooth and pale.
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Add a pinch of salt and pepper and mix.
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In a small sauce pan, heat oil o medium heat. Add mushrooms and cook until brown on each side, about 4-5 minutes on each side.
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Grill burgers to your desired doneness.
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Assemble burgers by spreading Aioli sauce on buns and stacking the hamburger patties, goat cheese, grilled mushrooms, arugula, and hot fries.
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Enjoy immediately!