Heat the oil in a pot over medium heat. Brown the turkey while breaking it up into smaller pieces using a spatula. Once it's halfway browned, add in the onion and garlic; continue cooking until the turkey is no longer pink. Season with salt and pepper.
Stir in the broth, butternut squash, and thyme. Cover with a lid and bring to a simmer for 15 minutes.
Stir in the kale and milk, return to a simmer for 10 minutes. Taste and adjust the seasoning with salt and pepper.
Slowly add the flour/water mixture to the simmering soup, continue to simmer until the soup thickens, about 5 minutes. Remove the thyme stems.
Ladle the soup into a bowl, garnish with parsley and serve with bread.