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In a small skillet, heat the butter and oil over medium-high heat. When the butter is almost melted, add the onion and salt. Cook, stirring occasionally, for 2 minutes, then stir in the garlic. When the onion is golden and soft, stir in the hot sauce, thyme, red pepper flakes, and pepper. Transfer to a large bowl and cool to room temperature.
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Combine the chicken with the cooled onions using slightly damp hands. You want it well mixed, but you don't want to squeeze it and make it tough. Form the mixture into 4 burgers. Use your thumb to dimple the center of each patty.
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Coat a large nonstick skillet with oil and heat over medium-high heat. Add the burgers and cook until browned, about 3 minutes, then carefully flip them. Cook until the other side is browned and the meat is cooked through, about 3 minutes longer. The burger will feel firm and the juices will run clear.
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Slather the spicy mayo on the buns. Divide the burgers among the bun bottoms and top with the slaw. Sandwich with the bun tops and serve immediately.