Season both sides of the halved chicken breasts with sea salt, freshly ground black pepper, and garlic powder.
Dredge each chicken piece in the all-purpose flour, shaking off excess flour.
Heat a large skillet over medium-high heat and add 3 tablespoons butter and the olive oil until the butter melts and foams.
Working in batches to avoid crowding, add the chicken and sauté about 3 minutes per side until golden and cooked through; transfer cooked chicken to a plate and repeat, adding more oil if needed.
In the same skillet, add the chopped shallot and mushrooms and sauté, stirring occasionally, until they begin to brown and are fragrant, about 2–3 minutes.
Add the baby kale and stir, cooking until the kale wilts.
Pour in the chicken stock and simmer until reduced by about half, roughly 3 minutes.
Reduce heat to low and stir in the remaining 3 tablespoons butter, rinsed capers, and fresh lemon juice; taste and season with additional salt and pepper if desired.
Return the cooked chicken to the skillet to warm briefly, then add the cooked fettuccine and toss to coat in the sauce until everything is heated through.
Serve immediately garnished with coarsely chopped parsley, if using.
