Toss portobellos with garlic, herbs, salt, pepper, vinegar, and 1/4 cup oil in a large bowl. Let stand at room temperature, tossing occasionally, for 1 hour.
Transfer portobellos to a plate. Pour marinade into a small saucepan, and bring to a boil. Reduce heat, and simmer 1 minute. Add stock. Raise heat, and bring to a boil. Reduce heat, and simmer until reduced by half, about 10 minutes. Stir in remaining tablespoon oil, and remove from heat.
Preheat grill to medium-high. (If you are using a charcoal grill, coals are ready when you can hold your hand 5 inches above grill for just 3 to 4 seconds.) Lightly brush top and bottom of polenta with oil, and place in center of grill. Cook, flipping once, until polenta is marked, crisp, and no longer sticks to grill, 6 to 8 minutes per side. Meanwhile, place portobellos on cooler part of grill, and cook, turning frequently to avoid burning, until tender, about 10 minutes.
Place arugula on a platter, and arrange polenta and portobellos on top. Drizzle about half the sauce over portobellos. Top with Parmigiano-Reggiano. Serve immediately with extra sauce on the side.