To prepare the sauce, toss the garlic cloves with the olive oil. Wrap the garlic in foil and bake until tender, about 20-30 minutes. When the garlic has cooled slightly, put it into food processor and purée with the mayonnaise, lemon juice, pepper, and salt. Set aside.
To prepare the fritters, heat the oil in a heavy-bottomed pot until it reaches 350°F. Line a baking or cooling rack with two layers of paper towels.
Place the onion, zucchini, and yellow squash in a mesh colander and press down to squeeze out most of the liquid. Set aside.
Combine the flour, cheese, salt, pepper, and garlic powder in a large bowl. In a separate bowl, beat the eggs until smooth and then stir in the cold beer. Combine the wet and dry ingredients and stir in the zucchini and onion.
Scoop out 1 Tbsp of the zucchini batter and drop it into the hot oil; repeat. Fry until golden brown. Use a slotted spoon to transfer the fritters to the paper towel-lined rack. Serve them immediately with the dipping sauce.
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