Combine the ground chuck, salt, and pepper in a medium bowl. Shape the mixture into 4 (1/2-inch-thick) patties.
Lightly drizzle both sides of the mushrooms with the oil. Sprinkle with salt and pepper.
Mix the mayonnaise, lemon juice, and garlic in a small bowl to blend. (The patties, mushrooms, and garlic mayonnaise can be prepared up to this point 1 day ahead. Cover them separately and refrigerate.)
Prepare the barbecue (medium-high heat). Grill the burgers until cooked to desired doneness, about 4 minutes per side for medium-rare. Grill the mushrooms until just tender, about 5 minutes per side. Grill the focaccia squares cut side down, until lightly toasted, about 2 minutes.
Spread the garlic mayonnaise over the roll bottoms. Cut the mushrooms into strips and arrange the mushroom strips over the mayonnaise. Place the burgers atop the mushrooms. Spread garlic mayonnaise on the cut side of the tops and arrange a handful of arugula on each. Cover the burger with the roll tops and serve.