Rinse chicken breast under cold water, removing any excess fat. Cut each breast in half. Pat chicken dry with a paper towel.
In a large Dutch oven or large, wide flameproof casserole heat oil over medium-high heat. When it just begins to smoke, add chicken pieces to fit into the pot without touching and saute until well browned on each side. Using tongs, transfer the chicken to a platter and set aside while you saute the remaining pieces.
When all the chicken pieces have been removed from the pot, add the onions and garlic and reduce the heat to medium low. Saute, stirring occasionally, until the onions are softened but not browned, about 5 minutes.
Return the chicken to the pot along with any accumulated juices, and add the wine. Increase heat to medium high and bring to a boil. Cook, uncovered, until the wine is reduced by half, about 5 minutes, then add the chicken stock, thyme, bay leaves and black pepper, and cayenne. Bring to a boil, then reduce the heat to low and cook at barely a simmer, partially covered, until the chicken is almost done, about 1 hour, uncovering occasionally to skim any foam.
Add the carrots and parsnip and continue simmering until chicken and vegetables are tender, about 15 minutes.
Meanwhile, prepare the batter for the dumplings. Combine the flour, baking powder, salt, chives and parsley in a medium-size bowl and whisk thoroughly to mix. Pour in the milk and stir to make a thick batter. Set aside.
Using a slotted spoon, gently transfer the chicken to a bowl or platter and set aside. Remove and discard the bay leaves. Taste the broth and add salt as necessary. Add the chilled beurre manie in walnut sized pieces and whisk until it is completely incorporated and the liquid is somewhat thickened. ( I like to shred chicken a bit at this point.) Return the chicken to the pot and bring to a simmer.
Drop the dumpling batter by tablespoonfuls directly onto the simmering stew. You should have approximately sixteen dumplings. The dumplings will expand quite a bit, so don't crowd them too much; leave about an inch between each. Cover tightly and simmer over low heat for exactly 15 minutes. DON'T BE TEMPTED TO PEEK! After 15 minutes the dumplings will be perfect.
Stir in your defrosted peas. Remove from heat and serve.