Pat the chicken dry with paper towels. Season with 1-1/2 tsp. salt and a few grinds of pepper, and let sit at room temperature while you heat the oil and prepare the batter.
Pour enough oil into a Dutch oven or other wide heavy-duty 6-quart pot to measure 3/4 inch. Heat the oil over medium-high heat until it registers 375°F on a deep fat/candy thermometer.
Meanwhile, in a medium bowl, whisk together the flour, 1 tsp. of the Old Bay, the baking powder, cayenne, 1 tsp. salt, and 1/2 tsp. pepper. In another medium bowl, whisk together the egg, 1/4 cup oil, and 1 cup water. Slowly whisk the flour mixture into the egg mixture until completely smooth.
Working in batches of 3 or 4, dip a chicken thigh into the batter, gently shake off any excess, and carefully lower it into the hot oil. Repeat with 2 or 3 more thighs, leaving about 1 inch of space between each thigh. Cook, turning once or twice, until golden-brown all over and an instant-read thermometer inserted into the thickest part of the thighs registers 165°F, 5 to 7 minutes. Transfer to a paper-towel-lined plate. Repeat with the remaining chicken, adjusting the heat as necessary.
While the chicken cooks, peel and chop the garlic, sprinkle with a generous pinch of salt, and mash to a paste with the side of a chef’s knife. Transfer to a small bowl and combine with the mayonnaise, lemon juice, and the remaining 3/4 tsp. Old Bay. Season to taste with more lemon juice or Old Bay and serve with the chicken.