Sauté onion: Heat olive oil in a large pot over medium heat and cook the diced onion until soft and translucent, about 4 minutes.
Add garlic: Stir in the minced garlic and cook for 30 seconds until fragrant, being careful not to let it brown.
Add carrots and broth: Add the shredded carrots and broth to the pot, then bring the mixture to a gentle simmer.
Thicken the base: Whisk the flour and water slurry into the simmering broth, stirring until it slightly thickens, about 2 minutes.
Cook the broccoli: Add the chopped broccoli florets and simmer until just tender but still bright green, 8 to 10 minutes.
Blend partially: Use an immersion blender to puree about half the soup, leaving plenty of broccoli pieces for texture.
Lower the heat: Reduce heat to low so the soup is just barely simmering before adding the dairy, which prevents curdling.
Add milk and yogurt: Stir in the milk and temper the Greek yogurt with a spoonful of hot broth before whisking it into the pot.
Melt in the cheese: Add the shredded cheddar a handful at a time, stirring until fully melted and smooth before adding more.
Season to taste: Stir in smoked paprika, nutmeg, black pepper and salt, tasting and adjusting as needed.
Serve warm: Ladle the soup into bowls while steaming hot and serve immediately with bread or crackers.
