Preheat an outdoor grill or cast-iron grill pan to medium-high.
Working on baking sheet, wrap each round of cheese in a radicchio leaf like an egg roll, and pinch them a little to help it keep their shape. Arrange them on 1 side of the baking sheet.
Place the mushroom caps on the other side of the baking sheet; drizzle with the olive oil, and season with salt and pepper. Grill the mushrooms until tender and juicy, turning once, about 6 minutes. Next, lightly drizzle the wrapped goat cheese with oil and grill until slightly charred and warm, turning occasionally, about 1 minute. Immediately remove and slice in half on the diagonal.
Slice the grilled mushroom caps into strips and add them to a bowl with the lemon juice and sherry wine vinegar. Toss to combine.
In a bowl, toss the lettuce mix with the mushrooms and arrange in 4 bowls. Arrange 2 goat cheese halves in the center of each. Sprinkle each salad with red pepper and walnuts, and serve garnished with lemon wedges.