Melt the butter in a large pot and saute the onion and carrots until softened, about 5 minutes over medium heat, stirring occasionally so the onions turn translucent.

Sprinkle in the all-purpose flour and cook for 1 minute, stirring constantly over medium heat until it smells slightly nutty and looks smooth (no dry flour streaks).

Gradually whisk in the chicken broth, keeping the mixture smooth; bring it up to a gentle simmer and watch for small bubbles at the edges.

Add the elbow macaroni and cook until al dente, about 8-10 minutes, stirring occasionally so the pasta doesn’t stick, until you can bite it and still feel a slight firmness.

Stir in the cooked shredded chicken and milk, then keep the soup at a low simmer for 2-3 minutes, until everything is evenly combined and steaming.

Reduce heat to low and gradually stir in the shredded cheddar cheese until melted and smooth, cooking just until the cheese disappears; the texture should turn glossy and thick enough to coat a spoon.

Season with salt and pepper to taste and serve hot, making sure the final flavor is balanced and the soup looks evenly creamy throughout.