Broccoli-Cheddar Soup

Heat oil in a large pot over medium-high heat. Add onion; cook, stirring occasionally, until softened and translucent, 4 to 5 minutes. Add flour, garlic and mustard; cook, stirring constantly, until the flour is toasted and the garlic is fragrant, about 1 minute. Slowly add broth, stirring constantly. Stir in milk. Bring to a simmer over medium-high heat, stirring often and scraping the bottom of the pot to release any browned bits. Reduce heat to medium to maintain a low simmer; add 7 cups broccoli. Simmer, stirring occasionally, until the broccoli is tender, about 8 minutes. Meanwhile, coarsely chop the remaining 3 cups broccoli; set aside.

Remove the pot from the heat. Using an immersion blender, process the mixture until almost smooth with some chunks remaining, 20 to 30 seconds. Add carrots and the chopped broccoli to the pureed mixture in the pot. Cook over medium heat, stirring occasionally, until the carrots and broccoli are just tender, about 6 minutes. Stir in salt and remove from heat. Add 2 1/2 cups Cheddar to the soup, 1 handful at a time, stirring to completely melt the cheese before adding the next handful. Stir in vinegar. Ladle the soup evenly into 8 bowls; top with chives and the remaining 1/2 cup Cheddar. Serve immediately.