Sweat onion, celery, herbs, and cayenne in butter in a covered Dutch oven over medium heat for 5-8 minutes until vegetables are softened. Stir in flour and cook 1 minute; don't worry about clumps.
Deglaze with wine, whisk in cicken broth, and simmer 15 minutes, or until thickened. Whisk in cream. Add chicken, peas, lemon juice, salt, and pepper; bring to a simmer. Stir carrots, squash and green beans into stew.
Meanwhile, make the dumpling dough. Warm buttermilk and 3 tablespoons butter in for the dumplings in a saucepan over low heat until butter melts (do not worry if the buttermilk curdles).
Combine cake flour, parsley, baking powder, and salt in a bowl. Stir buttermilk mixture into dry ingredients to form a wet dough.
Drop dough onto simmering stew with a cookie scoop or spoon. cover, transfer pot to the oven, and steam 15 minutes, or until a toothpick inserted in the dumplings comes out clean.