Dice the onion into 1/4-inch pieces and finely dice the carrots into similar-sized pieces—uniform sizes ensure even cooking. Measure out the flour, spices (garlic powder, paprika, salt, and black pepper), and have your chicken stock, half and half, and shredded cheddar cheese ready. In a large pot or Dutch oven, melt the butter over medium heat until foaming. Once melted, add the diced onion and cook for 3-4 minutes, stirring occasionally, until the onion becomes translucent and softens slightly. This builds the aromatic foundation for the soup.

Add the measured flour to the softened onion and stir constantly for 1-2 minutes to cook out the raw flour taste and create a roux—this thickens the soup and gives it body. Pour in 1 cup of the chicken stock while whisking continuously to break up any lumps and create a smooth, cohesive paste. The whisking prevents lumps from forming and ensures the flour is fully incorporated. Reduce the heat to medium-low and let this mixture cook for another minute while stirring.

Stir in the garlic powder, paprika, salt, and black pepper, mixing well to distribute the spices evenly. Slowly add the remaining 1 cup of chicken stock while stirring, then pour in the half and half. I like to add the half and half slowly and stir constantly to keep the mixture smooth and prevent any curdling—this technique keeps the soup silky and luxurious. Continue stirring until everything is well combined and the soup begins to thicken slightly.

Remove the pot from direct heat briefly and add the shredded sharp cheddar cheese in stages, stirring constantly after each addition until it melts completely. Sharp cheddar gives the soup much better flavor than mild varieties. Once the cheese is fully incorporated and the soup is smooth, add the frozen broccoli florets and diced carrots from Step 1. Stir to distribute the vegetables evenly throughout the soup. The frozen broccoli doesn't need thawing—it will cook perfectly in the hot soup.

Return the pot to medium-low heat and simmer gently for 25-30 minutes, stirring occasionally to ensure even cooking and prevent sticking on the bottom. The broccoli will soften, the carrots will become tender, and the flavors will meld together beautifully. Taste the soup and adjust seasoning as needed—I sometimes add a small pinch of extra salt or a crack of fresh pepper to brighten the flavors. Serve hot, and the soup will have that creamy, comforting Subway broccoli cheddar texture you're after.