Pat the chicken breasts dry, season both sides with sea salt, freshly ground black pepper, and the garlic powder. Dredge each piece in the 1/3 cup all-purpose flour, shaking off any excess.
Heat a large skillet over medium-high heat. Add 3 tablespoons of the unsalted butter and the 3 tablespoons extra-virgin olive oil; heat until the butter is melted and foaming.
Working in batches to avoid crowding, add enough chicken pieces to brown without overlapping. Sauté until golden brown and cooked through, about 3 minutes per side. Transfer cooked chicken to a plate and repeat with remaining pieces, adding more olive oil between batches if needed.
Reduce heat to medium. Add the finely chopped shallot and the baby bella mushrooms to the skillet and sauté, stirring occasionally, until the shallot and mushrooms begin to brown and are fragrant, about 2 to 3 minutes.
Add the baby kale and stir until wilted, about 1 to 2 minutes.
Pour in the 3/4 cup chicken stock and let the mixture simmer until the liquid is reduced by about half, approximately 3 minutes.
Turn the heat to low and stir in the remaining 3 tablespoons unsalted butter, the rinsed 1/4 cup brined capers, and the 1/4 cup fresh lemon juice. Season with additional sea salt and freshly ground black pepper as desired.
Add the 1 lb cooked fettuccine to the skillet and toss to coat the pasta in the sauce. Return the cooked chicken to the skillet (or place on top) and warm everything together for 1 to 2 minutes so the chicken and pasta are heated through and coated with sauce.
Serve immediately, garnishing with coarsely chopped parsley if desired.