Preheat broiler to low with oven rack in middle of oven. Toss together portobellos, garlic, onion, celery, carrots, and 1 tablespoon oil on a large rimmed baking sheet lined with parchment paper. Spread in a single layer. Broil in preheated oven until vegetables are charred in spots, about 20 minutes, rotating pan and flipping vegetables halfway through. Let cool 5 minutes. Squeeze roasted garlic out of skins; discard skins. Transfer garlic and vegetable mixture to a food processor. Pulse until finely chopped, about 4 times; set aside.

Heat 3 tablespoons olive oil in a large, deep skillet over medium-high. Add cremini mushrooms; cook until bottoms of mushrooms are browned, about 3 minutes. Stir and cook until browned on all sides, 5 to 8 minutes. Stir in tomato paste and oregano; cook, stirring constantly, about 1 minute. Add wine; cook, stirring constantly, until slightly thickened, about 1 minute. Stir in roasted vegetable mixture; season with salt and pepper. Remove from heat.

Bring a large pot of salted water to a boil over high. Cook pasta according to package directions for al dente. Drain, reserving 1½ cups cooking liquid. Transfer cooked pasta to skillet with sauce, and return to heat over medium. Add cheese and remaining 2 tablespoons oil. Stir in reserved cooking liquid, ½ cup at a time, until sauce is creamy, about 3 minutes. Season with salt to taste. Divide pasta among 4 warm bowls; top with grated cheese.