In a large Dutch oven, melt butter over medium heat. Add carrots, celery, and onion. Season with salt and pepper. Cook until softened, about 5 minutes.
Sprinkle in flour and stir for 2 minutes. Gradually add broth while scraping the bottom of the pot. Stir in pumpkin, cumin, pumpkin pie spice, and garlic.
Add chicken breasts, season again, and bring to a simmer. Cover and cook for 10–15 minutes, until chicken is cooked through.
Meanwhile, make the dumplings. In one bowl, whisk flour, baking powder, pumpkin pie spice, salt, and pepper. Add grated butter and toss.
In another bowl, mix pumpkin, milk, and egg. Combine with dry ingredients until just mixed—don’t overwork.
Remove chicken, shred it, and return to pot with squash or sweet potatoes and milk. Stir and bring back to a gentle simmer.
Scoop tablespoon-sized dumpling dough and drop into the soup. Cover and simmer for 8–10 minutes until fluffy and cooked through.
Stir in lemon juice, adjust seasoning, and serve warm.