Preheat the oven to 350 degrees F.
Coarsely grate the cheese. Line a baking sheet with a silpat mat or parchment paper. Sprinkle 1 tablespoon of the cheese on the mat to make a thin disk, about 2 inches wide. Repeat with the remaining cheese, spacing the disks evenly on the pan to make 8 fricos. Bake rotating the pan once during baking, until the cheese melts together and is just golden, about 12 minutes. Cool fricos on the pan.
Position a rack 6 inches from the broiler and preheat.
Slice the mushrooms as thin as possible. Toss with the lemon juice and season with the salt and pepper, to taste.
Brush the chicken breast with olive oil and mustard. Season with the salt and pepper. Place the chicken in a flameproof gratin or casserole dish and broil, turning as needed until an instant-read thermometer inserted into the breasts registers 160 degrees F, about 25 minutes. Transfer the chicken to a plate, reserve the pan. Add the broth to the pan, and stir with a wooden spoon to release the brown bits that cling to the pan. Reserve the pan drippings.
Whisk the balsamic vinegar, salt, and pepper, to taste, in a large bowl. Gradually whisk in the oil, and the pan drippings starting with a few drops and then adding the rest in a steady stream, to make a dressing.
Slice the chicken. Toss the spinach and mushrooms with the dressing and divide among 4 plates. Scatter the chicken over the salads, top with the fricos and serve.
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