Preheat a grill or grill pan to medium-high. Using a food processor, puree 1/4 cup water, the shallot, almonds, parsley, sun-dried tomatoes, vinegar, Worcestershire sauce and mustard. With the machine on, pour in 1/2 cup EVOO to form a smooth, thick dressing. Pour about two-thirds of the dressing into a large bowl. Add the mushroom caps and turn to coat.
Place the mushrooms on the grill, cover the grill (or cover the grill pan with foil) and cook, turning once, 7 minutes total. Brush the polenta slices with EVOO and season with salt and pepper. Grill until just heated through, about 4 minutes. Place the "burgers" on the polenta rounds, top with a slice of cheese and another polenta round and cook until the cheese melts, 1 minute more.
Toss the celery and arugula with the remaining dressing. Serve 2 "burgers" with the salad alongside.