Melt the butter in a large pot over medium heat. Add the diced onion, garlic, carrots, and celery. Sauté for about 5 minutes, or until the vegetables are softened.
Pour in the chicken broth and bring to a boil. Add the diced potatoes, thyme, parsley, salt, and pepper. Reduce the heat and let it simmer for 15-20 minutes, or until the potatoes are tender.
In a mixing bowl, combine the flour, egg, milk, and 1/4 teaspoon salt. Mix until a dough forms. Drop small spoonfuls of the dough into the simmering soup. Cook for an additional 10 minutes, or until the dumplings are cooked through.
Stir in the heavy cream and heat through. Adjust seasoning with more salt and pepper if needed. Serve hot.