For the mushrooms:
10 ounces brown or cremini mushrooms, tough stems removed, coarsely chopped
2 tablespoons butter, plus more for buttering the biscuits
1 medium shallot, minced
1 teaspoon chopped fresh thyme leaves
1/2 teaspoon fine salt
1/4 teaspoon freshly ground black pepper
1/2 cup sour cream
For the sandwiches:
4 biscuits, split in half
1/4 cup shredded Gruyère cheese
1 cup arugula
For the mushrooms: Place the mushrooms in the bowl of a food processor fitted with the blade attachment. Pulse until all the mushrooms are finely chopped but not puréed. (You might have to scrape down the sides of the bowl occasionally to re-distribute bigger pieces of mushroom.) Alternatively, you can finely chop the mushrooms by hand.
Melt the 2 tablespoons butter in a large skillet over medium heat. Add the shallot and sauté until translucent and fragrant, about 1 minute. Add the mushrooms, thyme, salt, and pepper. Cook, stirring often, until the mushrooms have softened and released their liquid, about 4 minutes.
Add the sour cream, increase the heat to medium-high, and simmer until some of the liquid is evaporated, 1 to 2 minutes. Take the pan off the heat and reserve.
For the sandwiches: Heat the broiler to high. Lightly butter the cut sides of the biscuits. Place the biscuit bottoms cut-side up on a foil-lined baking sheet. Top the biscuit bottoms with equal amounts of the mushroom gravy and Gruyère. Broil until the cheese is melted, 45 seconds to 1 minute, depending on your broiler.
Transfer the biscuit bottoms to 4 plates. Divide the arugula over the biscuit bottoms and close each sandwich with a biscuit top.