Melt the butter in a large Dutch oven or soup pot over medium-high heat. Add the ground beef and cook for 5-7 minutes, breaking it up, until browned.
Add the onion and carrots and cook for 4-5 minutes, until the vegetables begin to soften. Add the garlic and cook for 30 seconds.
Stir in the tomato paste, Worcestershire sauce, thyme, rosemary, salt, and black pepper. Cook for 1 minute.
Sprinkle the flour over the beef and veggies and cook, stirring constantly, for 1 minute.
Slowly pour in the beef broth, stirring as you go. Add the potatoes and bring the soup to a boil. Reduce the heat, cover the pot partially, and simmer for 15-18 minutes, or until the potatoes are tender.
Transfer 1 1/2 cups of the cooked potatoes and about 1/2 cup of the broth to a blender. Blend until smooth and stir the mixture back into the soup.
Add the half-and-half and cook over low heat for 2-3 minutes, stirring occasionally. Do not bring it to a hard boil.
Reduce the heat to low and gradually stir in the cheddar until completely melted. Do not let the soup boil.
Add the frozen peas and cook for 2-3 minutes, just until heated through.
Taste and adjust the seasoning as needed. Ladle into bowls, garnish with parsley, and serve.