In a large Dutch oven or pot, melt 2 tablespoons of butter over medium high heat. Add the garlic, onions, and Italian seasoning. Cook until softened, about 6-8 minutes, stirring occasionally. Stir in the kale, adding a little bit of water or broth to help it cook down for about 1-2 minutes.
Add the broccoli florets and the broth. Bring to a boil and then reduce to a simmer. Simmer for about 12-15 minutes or until the broccoli is softened, with the lid partially covered. If needed, add more broth.
Turn off the heat, and blend it using an immersion blender. Skim off any foam if needed. Note: If you don’t have an immersion blender, use a regular blender. Blend it in batches, being careful when handling hot liquids. Return the blended liquid to the pot.
Dissolve the all-purpose flour with the milk (the milk doesn’t need to be warmed up in advance). Add the milk and flour mixture to the pot. Turn the heat back on, and bring to a simmer. Simmer for about 5-7 minutes or until the soup is thickened, stirring often, uncovered. Season with salt and pepper to taste.
Check the consistency. If you want it to be thicker, dissolve 1 tablespoon of flour in about ¼ cup of milk and stir into the soup, simmering for a few more minutes to allow it to thicken. Continue doing so until desired consistency. If you want it to be thinner, simply add more milk or broth. Season with salt and pepper to taste one last time, if needed.
To serve, garnish shredded cheddar cheese. Serve immediately. Enjoy!