Heat butter in a large stockpot over medium heat. Cook and stir onion, celery, carrot, and garlic in hot butter until onion is translucent. Stir in potatoes, chicken stock, parsley, thyme, and cayenne pepper; bring to a boil. Cover, reduce heat, and simmer until potatoes are tender, about 15 minutes.

Meanwhile, mix together flour and egg in a bowl using a fork until mixture resembles grains of rice.

Stir milk and ham into the stockpot; bring to a simmer. Slowly sprinkle flour-egg mixture into soup, stirring constantly to prevent clumps. Simmer for 10 minutes, stirring occasionally.

Stir in broccoli and Cheddar cheese; cook until broccoli is tender and cheese is melted, about 5 minutes. Season with salt and pepper.