On a rimmed sheet pan, combine mushrooms, Brussels sprouts, and carrots.
Cut one onion half into 4 to 6 wedges; place wedges on pan. Roughly chop remaining onion half; and set aside.
Cut top off garlic head to expose cloves. Place on baking sheet.
Drizzle vegetables and top of garlic head with 4 tablespoons olive oil; sprinkle with ½ tsp. salt and pepper. Toss to coat (place garlic right-side up).
Bake at 400°F for 20 minutes. Flip vegetables; bake 15 minutes more. Set aside.
While vegetables are roasting, start the soup.
Heat remaining 3 Tbsp. olive oil in a Dutch oven over medium heat.
Add reserved chopped red onion; cook, stirring occasionally, until softened, about 5 minutes.
Add diced garlic and remaining ¼ tsp. salt; cook, stirring often, 1 minute.
Stir in potatoes, tomato paste, and stock. Increase heat to medium high. Using your hands, crush tomatoes into the pan; add puree from can.
Add bay leaves and thyme; bring to a simmer.
Reduce heat to medium; simmer until potatoes are tender when pierced with a fork, about 15 minutes.
Reduce heat to medium low. Squeeze roasted garlic cloves into pan; add remaining roasted
vegetables and green beans. Cook about 10 minutes more.
Garnish servings with fresh parsley, if desired.