In a large skillet, heat 1 teaspoon canola oil over medium heat. Stir in carrots and sauté for 2-3 minutes. Add zucchini and squash; cook until tender.
Heat 1 teaspoon canola oil in a separate large skillet over medium heat; stir in diced onions and sauté until golden brown - about 10 minutes. Add garlic and thyme, cook until fragrant for 1 to 2 minutes. Add dry barley to the skillet and lightly toast for 2 to 3 minutes.
Pour approximately 2 cups of warm broth into the barley mixture; reduce heat to low, and simmer, stirring occasionally, until broth is absorbed - about 5 to 10 minutes. Add the remaining broth, 1/2 cup at a time, allowing broth to absorb before each addition, until barley is tender - 15 to 20 minutes.
Stir in carrot mixture and Parmesan cheese. Cook until cheese melts and season with salt and black pepper.