Bring a large pot of salted water to a boil and cook the pasta until al dente according to package directions; reserve about 1/4 cup cooking water, then drain.
While the pasta cooks, cut the chicken breasts into bite-size pieces and season with salt and pepper.
Lightly dredge the seasoned chicken pieces in flour, shaking off any excess.
Heat the butter and olive oil in a large skillet over medium-high heat. Add the chicken and cook until browned and no longer pink inside, about 5–7 minutes; remove the chicken to a plate.
Remove the pan from the heat briefly, then add the minced garlic, chicken broth, lemon juice, Dijon mustard, and Italian seasoning. Return to medium heat and simmer for about 1 minute, scraping up any browned bits.
Add the cooked chicken back to the skillet, pour in the heavy cream, and stir to combine. Cook, stirring occasionally, until the sauce thickens slightly, about 3–5 minutes; reduce heat if the sauce bubbles vigorously.
Stir in the baby spinach and cook until wilted, about 1–2 minutes.
Toss the drained pasta into the skillet with the sauce, adding reserved pasta water a splash at a time if needed to loosen the sauce. Stir in the grated Parmesan and adjust seasoning with salt and pepper to taste before serving.