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Heat oil in a large Dutch oven over medium heat. Add onion; cook, stirring occasionally, until translucent, about 4 minutes. Add garlic; cook, stirring constantly, until fragrant, about 1 minute.
Add broth and salt; bring to a boil over medium-high heat. Add broccoli and carrots. Reduce heat to maintain a simmer; cook, stirring occasionally, until the vegetables are tender, about 8 minutes.
Meanwhile, whisk milk and cornstarch together in a liquid measuring cup; drizzle into the soup, stirring constantly. Cook, stirring constantly, until bubbling and thickened, about 2-4 minutes. Remove from heat; gradually add Cheddar, stirring until melted after each addition.