Position a rack 4 inches from the broiler element and heat the broiler to high.
In a medium bowl, mash the anchovy fillets with a fork or a wooden spoon until they form a paste. Stir in the vinegar, shallots, mustard, marjoram, 3/4 tsp. salt, and 1/8 tsp. pepper until well blended. Slowly whisk in the oil.
Wipe the mushrooms clean with a paper towel. Remove and discard the stems. Use a spoon to scrape out and discard the mushroom gills. Brush the mushroom caps with 4 Tbs. of the dressing, and sprinkle with 1/4 tsp. salt and 1/8 tsp. pepper. Set the caps, gill side up, on a rimmed baking sheet. Broil until the caps start to soften, 3 to 4 minutes. Flip them and continue to broil until they're well browned and tender, 3 to 4 minutes. Transfer the caps to a cutting board and cut into 1.4-inch slices.
In a large bowl, combine the baby greens and cherry tomatoes; add 6 Tbs. of the dressing and toss well to coat. To serve, set equal portions of the greens and tomatoes on dinner plates and arrange the mushroom slices on top. Drizzle each salad with about 1 Tbs. of the remaining dressing. Sprinkle with the crumbled goat cheese and serve.
Serve with toasted slices of sourdough bread rubbed with a garlic clove and the cut side of a tomato.