In a large heavy-bottomed pot (like a Dutch oven), heat the oil over medium-high heat.
Add onion, carrots, celery, and garlic. Cook for 5 minutes, until vegetables just start to soften. Add in the turkey and brown, breaking it up into smaller pieces. Stir in the flour and cook for 2 minutes.
Add chicken broth and thyme, stir to combine. Season with salt and pepper.
Bring to a low simmer for about 25 minutes, uncovered, while you make dumplings.
For the dumplings: In a large bowl, whisk together flour, baking powder, pepper, and salt. Make a well in the center of the flour mixture and pour in the milk.
Using a rubber spatula, stir together until a sticky dough ball forms. If your dough seems too dry, you can add extra tablespoon(s) of milk until it is sticky and fluffy.
Using a teaspoon, scoop the dough and drop directly into the simmering soup. Space them around the pot so they do not stick together.
Add in the peas, cover with a lid and simmer for 15 minutes, gently stirring. (cut 1 dumpling in half to make sure they are cooked through).
Taste and adjust the seasoning with salt and pepper if needed before serving.