Melt the butter in a large pot or dutch oven over medium-low heat.
Add the diced onion, carrots, garlic, and a pinch of salt and pepper to the pan and stir.
Cover the pot and allow the vegetables to cook for 6 minutes, stirring occasionally.
Turn the heat up to medium. Sprinkle the flour over the vegetables and cook, whisking constantly for 2 minutes.
Slowly whisk in the chicken stock until smooth
Add the broccoli florets to the pot. Simmer for 5-6 minutes, until broccoli is tender.
Meanwhile, in a medium bowl whisk together the heavy cream, sour cream, dijon mustard, and hot sauce.
Add 1 ladle full (about ½ cuof the broth from the soup to the cream mixture and whisk to combine.
Pour the heavy cream mix into the soup while whisking constantly. Taste the soup at this point, add salt & pepper to taste.
Remove the pot from the heat and stir in the shredded cheese in 3-4 handfuls until fully melted and smooth.
OPTIONAL FOR A SMOOTHER SOUP: Remove 1 ladle full (about 1 cup) of the soup from the pot into a bowl. Use a hand held immersion blender to carefully puree the soup to desired consistency. Return the reserved soup the the pot. OR, puree the soup in 2-3 batches in your blender until smooth.
INSTANT POT VERSION:
Turn the instant pot on to saute mode. Add the butter to the pot.
Once the butter is melted, add in the onions and carrots and cook, stirring occasionally for 5 minutes. Add the garlic and stir constantly for 30 seconds. Season with a pinch of salt and pepper to taste.
Sprinkle the flour into the pot and cook, stirring constantly for 2 minutes. Pour in the chicken broth slowly while whisking constantly. Turn the pot off.
Add the broccoli to the pot. Place the lid on the pot and set the valve to sealing. Cook on Manual, high pressure for 4 minutes. Allow the pressure to release naturally for 2 minutes before doing a quick release.
In a medium bowl whisk together the cream, sour cream, mustard and hot sauce. Whisk one ladle full of the soup into the sour cream mixture, then whisk into the soup. Add the shredded cheese to the soup in 3 additions, stirring well after each. Follow the directions above for blending if you want a smoother broccoli cheddar soup.
Serve immediately. Store leftover soup in the refrigerator for up to 2 days.
IMPORTANT RECIPE NOTES:
This soup is best when using freshly shredded cheese right off the block. I do not recommend using pre-shredded cheese for this soup, as it can be grainy.
You MUST remove your pan from the heat before you add your cheese. If you do not, the soup will curdle and become grainy.
Leftover soup, once cooled can be stored in an airtight container for up to 4 days in the refrigerator. This soup is cream/cheese based and does not freeze well.
Don't skip the dijon mustard and hot sauce unless you absolutely must. This soup does not taste like mustard, and it's not spicy. These ingredients simply enhance the cheese flavor.