Season the chicken pieces with salt and black pepper. Place them in a bowl and toss with ½ cup flour until they are lightly coated.
Heat 1 tablespoon of olive oil in a large Dutch oven or heavy pot over medium-high heat.
Add the chicken to the hot pot in a single layer. Cook for about 4 minutes, turning occasionally, until the chicken is lightly browned on the outside.
Remove the chicken from the pot and place it on a plate. Set aside.
Reduce the heat to medium. Add the butter and remaining tablespoon of olive oil to the pot.
Stir in the onion, carrots, and celery. Cook for about 5 minutes, stirring occasionally, until the vegetables begin to soften.
Add the garlic and cook for about 30 seconds, just until fragrant.
Pour in the chicken broth and stir well, scraping any browned bits from the bottom of the pot for extra flavor.
Return the browned chicken to the pot. Add the thyme and bay leaves, then bring the mixture to a gentle simmer.
Stir in the peas and heavy cream. Taste and adjust seasoning with salt and pepper if needed. Keep the mixture simmering while you prepare the dumplings.
In a mixing bowl, combine flour, baking powder, and salt for the dumplings.
Add the eggs and buttermilk, stirring until a soft dough forms. Do not overmix.
Drop spoonfuls of the dumpling batter into the simmering chicken mixture, leaving space between each one so they can expand while cooking.
Cover the pot with a lid and reduce the heat to low. Let the dumplings cook for about 15 minutes, until they become puffed and fully cooked.
Remove the bay leaves before serving. Spoon the chicken, dumplings, and creamy broth into bowls and serve warm.
Enjoy!