Set up the grill: Clean grates and set up for indirect heat. For gas, turn one burner to medium-low and leave others off. For charcoal, bank lit coals to one side. Aim for 325-350°F inside the grill with lid closed.
Brown the butter: Place butter in a small saucepan over medium heat. Swirl occasionally. It will melt, then foam, then the foam will subside. Watch closely — when milk solids at bottom turn deep amber and the butter smells nutty, pull it off heat. Pour into a heatproof bowl to stop cooking. The color should be like a well-toasted hazelnut.
Mix dry ingredients: In a large bowl, whisk together cake flour, baking powder, and salt. Set aside.
Mix wet ingredients: In a medium bowl, whisk brown sugar, buttermilk, eggs, and vanilla until smooth. Slowly stream in cooled browned butter (warm, not hot) while whisking constantly.
Combine batter: Pour wet ingredients into dry. Fold gently with a rubber spatula just until no streaks of flour remain. Do not overmix — a few small lumps are fine. Batter should be thick and glossy, like thick pancake batter.
Prepare skillet: Grease cast iron skillet generously with butter or oil. Pour batter into skillet and spread evenly. Sprinkle turbinado sugar over the top in an even layer.
Grill the cake: Place skillet on the indirect-heat side of the grill. Close lid. Bake for 45–55 minutes, rotating skillet halfway through. Doneness cues: cake pulls away from edges, top is golden brown and cracked slightly, toothpick inserted in center comes out with moist crumbs (not wet batter). Tap bottom of skillet — should sound hollow.
Rest and serve: Remove skillet from grill and place on wire rack. Cool in pan 15 minutes, then turn out onto rack to cool completely. Drizzle with honey-butter glaze if using. Serve warm or at room temperature.