Melt the butter in a large pot over medium heat. Add diced onions, celery, and carrots. Sauté for about 5 minutes until they soften, then stir in minced garlic and cook for one more minute.
Sprinkle the flour over the vegetables, stir well, and cook for 2 minutes until the mixture turns a light golden color.
Pour in the chicken broth and half-and-half gradually, stirring continuously to keep the mixture smooth.
Add dried thyme, mustard powder, salt, and pepper. Place the raw chicken breasts into the pot, bring to a gentle boil, and let it simmer for 10 to 15 minutes.
Remove the chicken from the pot, let it rest for 5 minutes, then dice or shred it and return to the soup.
Add potato gnocchi to the pot and simmer according to package instructions, about 5 minutes.
Stir in chopped spinach and red pepper flakes. Simmer until the spinach wilts, about 1 minute, then adjust seasoning if needed.
Serve warm for a comforting meal.