Heat a large nonstick skillet over medium-high heat. When very hot, add 1/2 tablespoon butter and half of the mushrooms; cook, stirring, until tender, about 5 minutes. Transfer to a bowl and repeat with 1/2 tablespoon more butter and the remaining mushrooms; transfer to the bowl. Add the herbs to the bowl and toss; season with salt and pepper. Wipe out the skillet.
Build the sandwiches: Spread each slice of bread with mustard and sandwich with the cheddar, ham, mushroom mixture and muenster.
Melt 1 tablespoon butter in the skillet over low heat. Add the sandwiches and cook, turning once and adding more butter if needed, until the bread is golden and the cheese melts, about 12 minutes.