Fresh bluefish, meaty and flavorful, is ideal for grilling. Pancetta slices, tied around the herb-stuffed fillets, keep the fish from sticking.
Heat grill. Season fish with salt and pepper. In a large skillet over medium heat, melt butter. Add celery, fennel, shallots, and garlic. Cook, stirring occasionally, until translucent, about 5 minutes.
Add mushrooms to skillet. Cook, stirring occasionally, until tender, about 4 minutes more. Stir in herbs and wine. Reduce heat to low, and simmer 2 minutes. Transfer mixture to a large bowl, and stir in the bread cubes. Season with salt and pepper, and let cool.
Lay first fillet, skin side up, on a clean work surface. Cover with half of the pancetta slices, and carefully flip over. Mound stuffing neatly on top of the fillet. Place second fillet on top, skin side up, and cover with remaining pancetta. Tie the stuffed fish securely with kitchen twine at 1-inch intervals. Replace any stuffing that falls out during the process, and make sure that all pancetta is kept in place by the twine.
Grill stuffed fish over medium heat until pancetta is charred, 5 to 10 minutes per side, using two spatulas to carefully turn fish. Cover grill, and continue cooking fish until just done throughout (fish will be opaque), about 15 minutes more. Transfer to a serving platter. Serve warm with lemon wedges.