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In a large pot, heat the oil and butter over medium-low heat until the butter melts. Add the onion and cook until it is very brown and soft, about 8 minutes (this develops big flavor so be patient). Increase the heat to medium. Add the carrots, and cook for 3 minutes, stirring occasionally. Stir in the garlic, salt, pepper, and thyme. Sprinkle the flour over the vegetables and mix to combine.
Cook until the flour disappears, about 2 minutes. Slowly stir in the broth. Add the milk, broccoli and cauliflower. Bring to a simmer. Place a lid over the pot, leaving it partially uncovered, and let simmer until the broccoli and cauliflower are tender, about 15 minutes, stirring occasionally.
Stir in the Dijon. With an immersion blender puree the soup until it is thick and creamy but still has some texture to it. (Alternatively, you can ladle a few scoops into a blender and puree a few batches. Be careful not to fill your blender more than halfway, as hot liquids like to splatter). Stir in the cheese. Continue stirring until the cheese is melted and the soup is creamy, 2 to 3 minutes. Taste the soup and season with additional salt and pepper to taste. Ladle the chowder into bowls, and serve warm, sprinkled with additional cheddar as desired.