Prepare outdoor grill with medium-hot coals, or heat gas grill to medium-high, or see Stovetop Method, below.
Pierce potatoes with fork. Place potatoes on piece of paper toweling in microwave oven. Microwave on 100% power 10 minutes or until partially cooked, but still very firm, turning over halfway through cooking.
When potatoes are cool enough to handle, cut each lengthwise into quarters.
Meanwhile, in a small skillet, melt the butter over medium heat. Add the walnuts, brown sugar, salt, black pepper, allspice and vinegar; cook, stirring, until the brown sugar is melted, about 2 minutes. Remove the skillet from the heat. Brush the cut sides of the potatoes with the brown sugar glaze, leaving the walnuts in the skillet.
Grill the potatoes, cut side down, uncovered, for 5 minutes. Turn the wedges, so the second cut side is facing down; grill 5 minutes. Turn the wedges, skin side down. Brush the cut sides with glaze. Grill for 5 minutes or until the potatoes are tender. Transfer the potatoes to a platter. Spoon remaining glaze with nuts over potato wedges.
Heat a stovetop grill pan over medium-high heat. Cook the sweet potato wedges as directed in step 5 on the grill pan instead of on the grill.