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Gather all ingredients.
Melt 1 tablespoon butter in a skillet over medium-high heat. Add onion; cook and stir until tender and translucent, about 5 minutes. Set aside.
Whisk ¼ cup melted butter and flour together in a large saucepan over medium-low heat. Continue to whisk and cook, adding 1 to 2 tablespoons of milk if necessary to keep the flour from burning, 3 to 4 minutes.
Gradually add milk while whisking constantly; stir in chicken stock and bring to a simmer.
Cook until mixture is thickened, about 15 to 20 minutes.
Add broccoli, carrots, celery, and sautéed onion; simmer until all the vegetables are tender, about 10 to 15 minutes.
Stir in Cheddar cheese until melted; season with salt and pepper to taste before serving.