Season the chicken with salt and pepper. Heat a large pot over medium-high heat and add olive oil and 2 tablespoons of butter. Once sizzling, add the chicken to the skillet and cook it on both sides until deeply golden brown and cooked through, about 5 to 6 minutes per side. Remove the chicken and set aside to a bowl, shredding once cool. I actually throw it in the bowl of my mixer and shred it with the paddle attachment!
Reduce the heat to medium and add the other 2 tablespoons of butter. Add the onions, carrots, celery garlic, rosemary and thyme. Toss in another pinch of salt and pepper, stirring well. Cook until softened, about 5 minutes.
While the vegetables cook, mix up the dumplings. Whisk together the flour, baking powder and salt in a large bowl. Combine the milk and the eggs, then stir into the flour until just mixed. Set aside for a few minutes.
Stir the 1/4 cup flour into the veggies with a wooden spoon until it coats the vegetables, then continues to stir until the flour cooks and turns golden, about 5 more minutes. Slowly stream in the chicken stock 1 cup at a time, stirring well the entire time so the mixture thickens. Add the chicken back into the pot. Increase the heat a bit and bring the mixture to a simmer, then cook while it bubbles for about 10 minutes. You want it to thicken up well! Stream in the heavy cream and stir well. Drop large spoonfuls of dough all around the top of the pot. I don’t worry about crowding the pot and cover the entire top of the “soup” mixture. You want the mixture to be bubbling for 10 to 15 minutes so it cooks the dumplings – after 10 minutes I usually gently flip the dumplings over.
Top the entire pot with fresh chopped parsley. This is a very dumpling heavy recipe, which I adore. Serve immediately! I love how this reheats – just add a bit of stock or milk when reheating!