Sauté the Vegetables: In a large pot (4-quart or larger), heat the olive oil over medium heat. Add the diced carrots, celery, and onions. Season lightly with salt and pepper. Sauté the vegetables until they begin to soften, about 5 minutes.
Make the Roux: Add the flour to the pot and cook for another minute, stirring constantly to incorporate the flour with the vegetables.
Add Broth and Vegetables: Combine the water and chicken base in a separate bowl. Add the mixture to the pot along with the broccoli and cauliflower florets. Stir to combine.
Simmer: Bring the soup to a boil, then reduce the heat to a simmer. Cover the pot with a lid and let it simmer for about 10 minutes, or until the vegetables are tender.
Mash the Vegetables: Turn off the heat. Using a potato masher, mash the vegetables in the soup until you reach your desired consistency.
Finish with Cream and Cheese: Add the half and half and shredded cheddar cheese to the pot. Stir until the cheese is fully melted and the soup is creamy. Taste and adjust the seasoning with salt and pepper if needed.
Serve: Serve the soup hot, garnished with extra cheese or fresh herbs if desired.