Heat a panini or sandwich press according to the manufacturer’s instructions. (Alternatively, heat a nonstick grill pan over medium-high heat.)
While the press is heating, put the basil, olives, nuts, and garlic in a food processor and process until finely chopped. With the motor running, add 6 Tbs. of the olive oil in a slow, steady stream through the feed tube and continue to process until thick and smooth. Season to taste with salt and pepper.
Brush the mushrooms with 2 Tbs. of the olive oil and sprinkle with salt and pepper. Put them on the press, pull the top down, and cook until softened and browned, 3 to 5 minutes (or cook in the grill pan, flipping once). Transfer to a plate and let cool slightly.
Spread the pesto on the bottom halves of the rolls. Put 2 mushrooms on each and then some cheese. Top the sandwiches with the other halves of the rolls. Brush both sides of the sandwiches with the remaining 2 tsp. oil.
Put the sandwiches on the press pesto side up, pull the top down, and cook until browned and crisp and the cheese is melted, 5 to 7 minutes, depending on how hot your machine is. (If using a grill pan, put a heavy pan on top of the sandwiches and cook, turning the sandwiches over once.) Carefully remove from the press and serve.