Slice the chicken breasts in half lengthwise.
Season the flour with salt and pepper in a shallow dish. Dip the chicken in flour, then eggs.
Heat butter and oil in a skillet over medium heat. Cook chicken until golden, about 3–4 minutes per side.
In the same skillet, add broth, wine, and lemon juice. Simmer the sauce for 2–3 minutes.
Transfer the chicken to the slow cooker and pour the sauce over it.
Cook on low for 3–4 hours or high for 1–2 hours, until the chicken reaches 165°F. Garnish with fresh parsley before serving.