On a plate, mix together the flour, Parmesan and 1 teaspoon each salt and pepper. Dredge each chicken breast in the mixture, evenly coating on both sides.
Heat a large pan over medium. Add olive oil and 2 tablespoons of butter to the pan and melt to combine. Add the chicken and cook until golden brown (but not cooked through), about 4 minutes on each side. Remove chicken from pan and set aside.
Add remaining 2 tablespoons of butter to the pan and let it melt. Add shallot, garlic and a pinch of salt and cook, stirring until the shallot is softened and the garlic is aromatic, about 2 minutes.
Add wine, broth, basil and oregano, and stir, scraping the browned bits from the bottom of the pan, until the liquid has reduced by about half, 3 to 4 minutes. Add the heavy cream and cream cheese and stir, allowing the cream cheese to soften and melt, until a thick sauce forms, about 6 minutes. Add baby spinach and stir until it is folded into the cream sauce and the spinach is beginning to wilt, about 1 minute.
Return the chicken breasts to the pan and simmer until the chicken is cooked through, 4 to 5 minutes. Remove from heat and serve immediately with freshly grated Parmesan on top.
