Dice the onion into 1/2-inch pieces, slice the carrots into 1/4-inch rounds, dice the celery into 1/4-inch pieces, mince the garlic cloves, and cut the broccoli into 1-inch florets. Set all vegetables aside. In a small bowl, whisk together the flour and smoked paprika—this prevents clumping when added to the hot pan. Blend the cottage cheese until completely smooth using a blender or food processor; set aside. Shred the cheddar cheese and have it ready. This organized prep ensures smooth cooking without interruptions.

Melt the butter in a large pot over medium heat. Add the diced onion, sliced carrots, and diced celery, stirring occasionally and cooking for about 10 minutes until the vegetables soften and begin to caramelize at the edges—this builds deep, savory flavor. Add the minced garlic and cook for 1 minute until fragrant, then immediately sprinkle the flour-paprika mixture over the vegetables and stir constantly for 1 minute to toast the flour and create a roux. This brief cooking develops the flour's nutty flavor and ensures a smooth, lump-free soup.

Pour the vegetable broth into the pot, stirring to incorporate any browned bits from the bottom—this adds incredible flavor. Add the broccoli florets and bring the mixture to a gentle simmer. Cook for 8–9 minutes until the broccoli is very tender, which allows it to break down slightly and thicken the soup naturally while still holding some texture. I find that slightly overcooked broccoli works better here since we'll be blending part of the soup anyway, creating a creamier consistency.

Reduce the heat to low and pour in the milk, stirring gently to combine. Let the soup warm through for 2–3 minutes, then add the blended cottage cheese and stir until fully incorporated. Finally, add the shredded cheddar cheese in handfuls, stirring after each addition until completely melted. Keep the heat low—this prevents the dairy from scorching and ensures the cheeses melt smoothly into a silky sauce.

Using an immersion blender, blend about 1 cup of soup directly in the pot by submerging the blender head and blending for 30–45 seconds—this creates a creamy, velvety texture while leaving some broccoli florets intact for pleasant texture contrast. Alternatively, carefully transfer 1 cup of soup to a blender, blend until smooth, and return it to the pot. Taste the soup and add salt and freshly ground black pepper to your preference. I like to add the salt gradually and taste between additions, as the vegetable broth and cheeses already contribute saltiness.