Pat the chicken breasts dry with paper towels and season them evenly on both sides with kosher salt and black pepper.
Heat 2 tablespoons of the butter in a large non-stick skillet over medium heat. Once the butter has melted and begins to bubble, add the seasoned chicken breasts to the skillet. Cook for 8 to 10 minutes, flipping halfway through, until the chicken is golden brown on both sides and reaches an internal temperature of 165°F. (Exact cooking time will depend on the thickness of the chicken.)
Transfer the cooked chicken to a plate and loosely cover with foil to keep warm.
In the same skillet, melt the remaining 3 tablespoons of butter. Add the diced shallot and cook, stirring frequently, for 3 to 5 minutes until softened and translucent. Stir in the minced garlic and cook for another 1 minute until fragrant.
Sprinkle the flour over the shallot and garlic mixture, whisking continuously for about 1 minute until the flour is fully combined and begins to develop a light golden color.
Slowly pour in the chicken broth while whisking constantly to incorporate and loosen any browned bits stuck to the bottom of the skillet. Simmer the mixture for 3 to 5 minutes, or until the sauce has thickened slightly.
Add the half-and-half along with the uncooked potato gnocchi. Stir everything together so the gnocchi are well coated and mostly submerged in the sauce. Let it cook for 5 to 7 minutes, stirring occasionally, until the gnocchi are tender and fully cooked.
Fold in the fresh baby spinach and finely grated Asiago cheese. Continue to cook for another 2 to 3 minutes, stirring gently, until the spinach is wilted and the cheese has melted into a smooth, creamy sauce.
Remove the skillet from the heat and return the cooked chicken breasts to the pan, placing them on top of the gnocchi and sauce.
Sprinkle with chopped fresh parsley and serve immediately while hot.