In a large, heavy pot (I used a 5 1/2 quart pot), melt the butter over medium heat and stir in chopped shallots and carrots. Sauté for about 2 minutes until they are fragrant and more tender, stirring often.
Add the flour to fully coat the vegetables, then pour in about 1/2 cup of broth and whisk till the flour gets incorporated and smooth. Add the rest of the broth as well as the milk. Bring to a low simmer for 8-10 minutes, allowing it to thicken. Do not start to boil, otherwise your milk might curdle (*especially if using a low fat option instead).
Add the broccoli florets and cook for another 5 minutes until softened. Off the heat, stir in Boursin cheese until smooth and creamy. Taste and adjust any seasonings, like salt and pepper if needed.
Video
Notes
This soup will freeze well up to 4 months and can be reheated in the microwave or stove top.
I recommend using whole milk to give the soup extra creaminess and also the higher fat content makes it less likely to curdle, however you can use any milk substitute you'd like - just be aware!
Dairy free - Use dairy free Boursin, olive oil instead of butter, and oat milk.
Gluten free - Use arrowroot powder or gluten free 1:1 flour to thicken the soup.