Slice each chicken breast in half horizontally with a sharp knife to make four thin cutlets. If needed, place between plastic wrap and lightly pound to an even thickness (about 1/2 inch). Pat the cutlets dry and season both sides with kosher salt and black pepper.
Heat a large skillet over medium-high heat. Add 2 tablespoons olive oil and heat until shimmering. Add the chicken cutlets (work in batches if they don't fit without crowding) and cook until golden brown and cooked through, about 4–6 minutes per side (internal temperature 165°F). Transfer cooked chicken to a plate and tent loosely with foil to rest.
Reduce the skillet heat to medium and add 1 tablespoon butter. When the butter melts, add the chopped shallot and cook, stirring, until softened, about 2–3 minutes. Add the minced garlic and cook until fragrant, about 30–60 seconds.
Add the 2 cups roughly chopped spinach and the drained, roughly chopped artichoke hearts to the skillet. Cook, stirring, until the spinach is wilted, about 1–2 minutes.
Pour in 3/4 cup chicken broth, scraping up any browned bits from the bottom of the pan. Add 1 cup heavy cream and 1/4 cup freshly grated Parmesan cheese, stirring to combine. Reduce the heat to low and simmer, stirring occasionally, until the sauce thickens slightly, about 2 minutes.
Stir in 1 tablespoon lemon juice and 1/4 teaspoon crushed red pepper flakes. Taste and add additional kosher salt and black pepper if needed.
Return the rested chicken cutlets to the skillet, spooning sauce over them. Simmer on low for 2–3 minutes to warm the chicken and meld flavors.
Transfer the chicken to plates, spoon the sauce over the top, and garnish with extra Parmesan cheese and freshly chopped basil. Serve immediately.