Pat chicken breasts dry, season both sides with sea salt, freshly ground black pepper and 1 tablespoon garlic powder. Dredge each piece in 1/3 cup all-purpose flour, shaking off any excess flour.
Heat a large skillet over medium-high heat. Add 3 tablespoons unsalted butter and 3 tablespoons extra-virgin olive oil and heat until the butter is melted and the fat is shimmering.
Working in batches to avoid crowding, add the dredged chicken to the skillet and sauté until golden brown and cooked through, about 3 minutes per side (internal temperature 165°F / 74°C or until no longer pink in the center). Transfer cooked chicken to a plate and keep warm; add a little more olive oil to the pan between batches if it becomes dry.
While the chicken rests, slice the 8 oz. baby bella mushrooms if desired and finely chop the shallot (if not already chopped). Add the shallot and mushrooms to the same skillet and sauté over medium-high heat, stirring occasionally, until the shallot is softened and the mushrooms begin to brown and become fragrant, about 2 to 3 minutes.
Add the 6 oz. baby kale to the skillet, stir to combine, and cook until the kale is wilted, about 1 to 2 minutes.
Pour in 3/4 cup chicken stock and bring to a simmer. Let the liquid reduce by about half, approximately 3 minutes.
Reduce heat to low and stir in the remaining 3 tablespoons unsalted butter, 1/4 cup brined capers (rinsed), and 1/4 cup fresh lemon juice. Stir until the butter is melted and the sauce is smooth.
Return the cooked chicken to the skillet along with any accumulated juices on the plate. Add 1 lb. cooked fettuccine pasta and gently toss or stir to coat the pasta and chicken evenly with the sauce; heat just until everything is warmed through, about 1 to 2 minutes.
Taste and adjust seasoning with sea salt and freshly ground black pepper as needed. Serve immediately, garnished with coarsely chopped parsley if desired.