Slice each of the 2 large chicken breasts in half lengthwise to make 4 cutlets. If any pieces are thicker at one end, gently pound those areas to an even 1/2-inch thickness. Pat the cutlets dry with paper towels.
Season both sides of each cutlet with 1 teaspoon kosher salt, 1/2 teaspoon garlic powder, and 1/4 teaspoon fresh black pepper. Dredge each cutlet in the 1/4 cup flour, shaking off any excess.
Heat 1 tablespoon extra virgin olive oil in a large deep skillet over medium heat until shimmering. Add the chicken (in a single layer; cook in batches if necessary) and cook about 3 to 4 minutes per side, until golden-brown and cooked through. (Internal temperature should reach 165°F if using a thermometer.) Transfer the cooked chicken to a plate and set aside. Wipe out the skillet with a paper towel.
Reduce the heat to medium-low and add the remaining 1/2 tablespoon olive oil to the skillet. Add 1/3 cup chopped shallots and 2 cloves minced garlic; sauté about 2–3 minutes until fragrant and the shallots are translucent.
Add 1/3 cup diced red bell pepper and 9 ounces fresh baby spinach. Cover or partially cover the skillet and cook, stirring 2–3 times, until the spinach is fully wilted, about 4–5 minutes.
Reduce heat to low. Add 2 ounces cream cheese, 3/4 cup chicken broth, 3 tablespoons half & half, the remaining 1/4 teaspoon kosher salt, and most of the 1/4 cup grated Parmesan cheese (reserve a little of the Parmesan for serving). Stir continuously until the cream cheese melts and the sauce is smooth and combined.
Return the cooked chicken and any accumulated juices to the skillet. Spoon sauce and spinach over the chicken and simmer on low for 2–3 minutes, until the chicken is heated through and the flavors meld.
Serve the chicken topped with the reserved grated Parmesan.