Preheat oven to 400 degrees F (200 degrees C). Brush a jelly roll pan with a thin coat of olive oil.
Arrange carrots, tomatoes, onion, squash, potatoes, and garlic on the prepared pan. Drizzle remaining olive oil over the vegetables; season with salt and pepper. Turn the vegetables with a spoon to help coat evenly with the oil.
Roast in preheated oven until browned and tender, turning occasionally, 20 to 30 minutes. Set vegetables aside.
While the vegetables roast, place a large skillet over medium heat. Crumble sweet Italian sausage and hot Italian sausage into the skillet. Pour 1/4 cup vegetable broth over the sausage. Cook, breaking the sausage apart with a spoon as it cooks, until the sausage is cooked through and no longer pink, 7 to 9 minutes. Transfer sausage to a paper towel-lined plate to drain.
Cut squash and carrots into 1/2-inch pieces; set aside.
Peel garlic cloves; place in a blender with roasted tomatoes and onion and blend until smooth. Pour the mixture into a large pot.
Pour 1/2 cup broth onto the jelly roll pan. Scrape any browned bits from the surface of the pan with a wooden spoon or spatula; pour broth and browned bits into the pot with the blended tomato mixture.
Pour remaining vegetable broth into the pot. Add kale, thyme, and bay leaf to the mixture; bring to a boil. Reduce heat to medium-low and simmer until the kale is tender, about 30 minutes. Add sausage, carrots, potatoes, squash, garbanzo beans, and kidney beans to the soup; continue cooking until hot, about 10 minutes. Thin the soup with more vegetable broth as desired. Season with salt and black pepper. Discard thyme sprigs and bay leaf to serve.
