Dice the onion into 1/2-inch pieces and finely dice the carrots into 1/4-inch pieces, then set aside. In a large pot or Dutch oven, melt the olive oil and butter together over medium heat. Once foaming, add the diced onion and carrots, stirring occasionally until softened and the onion becomes translucent, about 5 minutes. This creates a flavorful base that will infuse the entire soup.

Sprinkle the flour over the softened aromatics and stir constantly for about 2 minutes to cook out the raw flour taste and create a roux—this will help thicken the soup and give it body. Slowly pour in the vegetable broth while stirring to avoid lumps, then add the milk and stir until fully combined and smooth. Add the bay leaves and bring the mixture to a gentle simmer.

Cut the broccoli into 1-inch florets and add them to the simmering soup along with the garlic powder, paprika, cayenne pepper, and freshly grated nutmeg. Season with kosher salt and black pepper to taste, stirring well. Simmer uncovered for about 10 minutes until the broccoli is just tender but still holds its shape. I like to add the spices early so they bloom and develop deeper flavor throughout the cooking time.

Remove the pot from heat and carefully remove the bay leaves. Using an immersion blender, blend approximately half of the soup directly in the pot, creating a partially chunky, partially creamy texture—this technique gives you the best of both worlds: body and some broccoli texture. Return the pot to low heat and stir in the shredded cheddar cheese and fresh thyme until the cheese is completely melted and the soup is smooth and creamy. Taste and adjust seasonings as needed.

While the soup finishes cooking, cook the egg noodles according to package directions until al dente, then drain well. Divide the cooked noodles from Step 5 among serving bowls, then ladle the hot broccoli cheddar soup from Step 4 over top. I prefer to keep the noodles separate during cooking so they don't absorb all the broth and become mushy—this keeps them tender and lets each spoonful have the right ratio of noodles to soup.