Heat olive oil in large Dutch oven over medium-low heat.
Add carrots, onions, broccoli, and salt, then cook on medium-low for 12-14 minutes.
Add garlic and stir until fragrant, about 1 minute.
Stir in broth, half-and-half, mustard, sugar, hot sauce, paprika, and turmeric. Turn heat to medium-high.
Add cheeses and stir until melted. Bring to boil, then return heat to simmer, stirring frequently.
Combine cornstarch and 1 1⁄2 tablespoons water to make slurry, add to soup, and simmer until thickened.