Preheat oven to 400°F. Pierce each yam in several places with fork. Place directly on oven rack and roast until just tender, 40-50 minutes, depending on size of yams.
Transfer roasted yams to work surface. Using scissors, cut off ends of each yam, then cut skin on 1 long side and peel off in 1 piece, being careful not to break warm yams. Let yams cool to room temperature. Tear or cut skins into long ribbons, about 1 inch wide.
Cut yams crosswise into 1/2" thick rounds. Arrange rounds, slightly overlapping, in 13" x 9" x 2" ovenproof dish. Cover yams with aluminum foil to keep moist.
Melt butter in large skillet over medium heat until it browns, stirring occasionally, 8-10 minutes. Carefully pour all of the butter & browned milk solids into a small bowl.
Blend shallot, garlic, and salt in food processor until smooth, occasionally scraping down sides of bowl. Add mustard, then red wine vinegar. Gradually drizzle in butter, including browned milk solids, while processor is running. Season to taste with salt & pepper, if desired do ahead Vinaigrette can be made 1 hour ahead. Let stand at room temperature. Blend again 5 seconds before using.
Pour enough vegetable oil into large deep skillet to reach 1 inch depth. Attach deep-fry thermometer to side of skillet and heat oil to 350°F. Add yam skins in batches (oil will bubble up) and fry until skins are crisp and brown, stirring occasionally, about 3 minutes. Using slotted spoon, carefully transfer yam skins to paper towels to drain. Sprinkle skins with coarse salt and pepper.
Spoon vinaigrette over potatoes. Scatter crisp skins over and serve.