First, prep the chicken. Trim any excess fat from the chicken breast and cut into bite-sized chunks. Add to a bowl, sprinkle with ½ teaspoon kosher salt and ½ teaspoon freshly ground black pepper, then add the flour and toss with a spatula or spoon to coat the chicken.
Heat 1 tablespoon of the canola oil in a large skillet with 1 tablespoon of butter over medium-high heat. Add half of the chicken and cook for 2-3 minutes per side. Transfer to a platter or sheet pan and cover with foil and wipe out any burned bits. Add more oil if needed and repeat with the remaining chicken.
Meanwhile, add the gnocchi to a large sheet pan. Drizzle with olive oil and toss with ½ teaspoon of kosher salt. Spread the gnocchi into an even layer. Roast in the oven for about 5-7 minutes, tossing halfway through.
While the gnocchi roasts, melt 3 tablespoons of the butter in the same skillet used for the chicken and add the mushrooms and shallot. Sauté over medium heat, season with the remaining ½ teaspoons of kosher salt and black pepper, and stir occasionally. Cook until the mushrooms and shallots have softened, about 5 minutes. Add the marsala wine and the chicken stock. Bring to a boil and stir in the remaining butter, then reduce to medium heat and cook for 2-3 minutes before adding the chicken and transferring the hot gnocchi from the sheet pan to the skillet. Continue cooking the gnocchi in the sauce until the sauce is reduced by half. Add more salt or pepper to taste and sprinkle with fresh parsley.