Boil peeled and cubed russet potatoes in salted water until tender, about 15 minutes. Drain, then mash with a splash of whole milk and set aside.
Season the chicken breasts evenly with salt and black pepper.
Heat the olive oil and butter in a large skillet over medium-high heat. Add the chicken and sear for 5-7 minutes per side until golden brown and cooked through (165°F internal temperature). Transfer to a plate.
In the same skillet, reduce heat to medium and add the minced shallot. Sauté for 2 minutes until softened.
Pour in the chicken broth, scraping up any browned bits from the bottom of the pan. Simmer for 1 minute.
Stir in the Boursin cheese until completely melted and the sauce is smooth.
Return the chicken to the skillet, spooning the sauce over the top. Simmer for 2 minutes until heated through.
Steam the green peas in the microwave or on the stovetop until tender.
Serve the chicken and sauce hot alongside the mashed potatoes and steamed peas.