Rinse and pat chicken dry with paper towels and slice the breasts horizontally. Season with salt and pepper. Heat one tablespoon of the oil in a large cast iron skillet over medium-high heat until just smoking. Cook chicken until golden, about 3 to 5 minutes per side. Transfer to a plate.
Add the remaining oil to the skillet and heat until shimmering. Cook potatoes, cut-side down, until golden brown, about 4 minutes.
Add the mushrooms and cook for three minutes.
Stir in garlic, rosemary, oregano, and ½ teaspoon salt and cook until fragrant, about 30 seconds. Add wine and broth, scraping up any browned bits, and bring to a boil. Return browned chicken to skillet on top of potatoes. Reduce heat to medium-low and simmer, covered, until chicken is cooked through and potatoes are tender, about 12 minutes. Using a slotted spoon, transfer chicken and potatoes to a serving platter; tent with foil. If the potatoes are not tender, place them on a cooky sheet lined with foils and cook in the oven at 400 for seven minutes.
Increase heat to medium-high and cook, uncovered, until sauce is reduced to 1 cup, about 5 minutes. Stir in peas and cook until heated, about 1 minute. Whisk in butter and lemon juice, and season with salt and pepper. Pour sauce over chicken and potatoes. Serve.