Step 1: Set up your grill for indirect grilling and preheat to medium (350ºF).
Step 2: Arrange the brioche chunks in a single layer in a large disposable aluminum foil pan. Place the pan on the grill grate away from the heat and cover the grill. For even more smoke flavor, toss a couple handfuls of soaked and drained wood chips on the coals. Indirect grill the brioche, stirring occasionally so the chunks brown evenly, until toasted and golden brown, about 15 minutes. Set the pan of brioche aside to cool.
Step 3: Meanwhile, trim the ends off the mushroom stems; remove and discard the stems if using shiitakes. Wipe the mushrooms clean with a damp paper towel. Cut large mushrooms into 1/4-inch slices; leave small ones whole.
Step 4: Melt 3 tablespoons of the butter in the cast-iron skillet on the stovetop over medium-high heat. Add the scallions, celery, if using, and sage and cook, stirring often, until golden brown, 4 minutes. Add the mushrooms and pecans. Increase the heat to high and cook, stirring often, until the mushrooms brown and all their liquid evaporates, 5 minutes. Add the Cognac, if using, and boil until only 1 tablespoon of liquid remains. Remove from the heat and let cool slightly.
Step 5: Crack the eggs into a large bowl and whisk until smooth. Whisk in the cream. Stir in the mushrooms, then the brioche cubes. Grate in the nutmeg and stir in the salt and pepper; the mixture should be highly seasoned. Spoon it back into the skillet and top with the remaining 2 tablespoons butter, cut into thin slices. (The pudding can be prepared several hours ahead to this stage; cover with plastic wrap or aluminum foil and refrigerate, if you have room. But the texture will be better if you cook it right away).
Step 6: If you shut down the grill after toasting the brioche, fire it up again for indirect grilling and preheat to 350ºF. Toss wood chunks or chips on the coals following the manufacturer’s instructions. Cover the grill and smoke-roast the pudding until puffed and browned on top and cooked through (a skewer inserted into the center should come out clean), 45 minutes to an hour. Serve the pudding right from the skillet.