Pat the chicken breasts completely dry with paper towels (dry surface = better sear). Season with salt and pepper, then dredge lightly in the flour, shaking off the excess.
Heat the olive oil and 1 tbsp of the butter in a large skillet over medium-high heat until the butter foams.
Sear the chicken 5 to 6 minutes per side, until deep golden and the internal temperature reads 165°F on a meat thermometer. Transfer to a plate and tent with foil.
Reduce the heat to medium. Add the remaining 2 tbsp butter and the garlic; cook 30 seconds until fragrant (don’t let it brown).
Stir in the honey, soy sauce, and vinegar. Simmer 2 minutes, stirring, until the sauce thickens enough to coat a spoon.
Return the chicken to the pan and turn to coat in the glaze, 1 to 2 minutes per side. Spoon extra sauce over the top.
Rest 5 minutes, then slice and top with green onions.