0 Prep the vegetables and get all the ingredients in order. Chop everything before you start cooking, and have everything out so you don't have to leave the stovetop.
1 Warm the olive oil over medium-high heat in a large Dutch oven or soup pot.
2 Add the chopped onions to the pot. Season with salt and pepper, and saute until soft, about 5-7 minutes. Stir constantly so the onions don't begin to brown.
3 Add the garlic and thyme, then cook until fragrant, about 1-2 minutes.
4 Sprinkle in the flour, stirring until the onions are coated. Cook for 2-3 minutes so that the flour can get a little color to it before proceeding. This will help take away that raw flour flavor.
5 Slowly pour in the vegetable stock, stirring constantly to avoid lumps. Scrape the bottom of the Dutch oven as you do this to pull up any browned bits left behind from the onion.
6 Bring to the soup to boil, then reduce the heat to the lowest setting on the stovetop. Simmer until thickened, about 5 minutes.
7 Add the chopped broccoli and cover the pot. Cook until tender, about 10 minutes, while opening the lid to stir occasionally.
8 Remove the pot from the heat. Puree with a hand-held immersion blender or a blender, then return to the pot.
9 Add the milk and cheese. Stir until incorporated and melted. This might take a few minutes, depending on how hot your mixture still is.
10 Serve warm, and enjoy!
