Heat grill to medium-high. Once it is hot, clean the grill grate with a wire brush. Just before grilling, oil the grill grate.
Combine the cornichons, capers, vinegar, 1 tablespoon of the oil, and ¼ teaspoon each salt and pepper in a small bowl.
Drizzle the mushrooms with the remaining 2 tablespoons of oil, ½ teaspoon salt, and ¼ teaspoon pepper.
Grill the mushrooms, covered, until tender, 5 to 10 minutes per side. Turn 4 mushrooms stem-side up. Dividing evenly, place the fontina on these 4 mushrooms and top with the remaining mushrooms, stem-side down. Grill, covered, until the fontina is melted, 1 to 2 minutes more.
Hollow out the rolls by scooping out the soft core. Stack the lettuce, burgers, and relish between the rolls.