Cook broccolini in a pot of boiling salted water just until tender, about 3 minutes. Drain and transfer to a bowl of ice water; let cool. Drain and pat dry with paper towels.
Heat butter in a medium skillet over medium until foaming. Add broccolini, lemon zest, and red pepper flakes; season with salt and pepper. Cook, tossing, just until broccolini is warmed through and coated in butter.
Heat a grill pan over medium-high until hot but not smoking. Pat black bass fillets dry; this will help keep the fish from sticking to the pan while cooking. Place flour on a plate and season generously with salt and pepper (you are not seasoning the fish itself). Working one at a time, dredge fillets in flour, shaking off excess. Spread butter over flesh side of fillets as if buttering a slice of bread.
Add fillets, flesh side down, to pan; they should sizzle upon contact. Cook just until grill marks appear and butter is browned, about 1 minute. Turn and cook, pressing gently so entire skin side is in contact with the pan's ridges, until skin is browned and crisp and fillets are cooked through, about 4 minutes longer.
Serve black bass fillets with broccolini and flavored butters, which will soften on contact with the warm fish.
