Pound chicken to 1/4-inch thick between plastic wrap or in large food storage bag. Season with salt and pepper and dress with fennel, oregon and chili flakes.
Heat a tablespoon of olive oil in large, cast-iron skillet over medium-high heat. Add chicken skin-side down and crisp skin, pressing down with spatula while skin browns, about 5 minutes. Turn chicken and cook 5 minutes more pressing down with a spatula to keep the breasts from shrinking up too much. Remove to a warm platter and cover with loose foil.
Add remaining 2 tablespoons olive oil to the skillet and place potatoes in cut-side down. Brown 6-7 minutes, shake pan and add shallots, garlic, thyme, peas, salt and pepper. Cook 4-5 minutes more, add wine to deglaze then stock, lemon juice and melt butter into sauce. Add parsley to finish, arrange crispy chicken on top and serve from the skillet at the table.