Step 1: Set up your grill for direct grilling and preheat to medium high. Grill the brioche slices until browned on both sides, 1 to 2 minutes per side. Let cool on a wire rack.

Step 2: Meanwhile, make the mushroom hash: Trim ends off the mushroom stems. Wipe the mushrooms clean with a damp paper towel and thinly slice. Melt 2 tablespoons butter in a large skillet (preferably the 12-inch skillet in which you’ll cook the pudding). Add the shallots and cook over medium heat until lightly browned, 3 minutes. Add the mushrooms and sage and cook over high heat until the mushroom liquid evaporates and the hash is lightly browned.

Step 3: Place the eggs in a large mixing bowl and beat until smooth. Beat in the half and half. Stir in the mushroom hash, followed by the bread cubes. Correct the seasoning, adding salt and pepper to taste: the mixture should be highly seasoned. Re-grease the skillet and spoon the pudding into it.

Step 4: Set up your grill for indirect grilling and preheat to 350 degrees. Add wood chips or chunks to the fire.

Step 5: Smoke-roast the pudding until browned on top and cooked through, about 45 minutes. To test for doneness, insert a metal skewer—it should come out clean.

Step 6: Let cool for 5 minutes, then dig in.