Whisk together the flour, 1 tablespoon salt and a few grinds of pepper in a pie plate or shallow dish. Coat both sides of the chicken breasts in the flour. Shake off any excess and set aside.
Heat 3 tablespoons of oil in a large, heavy-bottomed skillet over medium-high heat. Add the chicken to the skillet and cook, undisturbed, until deeply golden, about 3 minutes. Turn the chicken and cook, undisturbed, until the other side is deeply golden brown, about 3 minutes. Remove the chicken to a plate. Add the remaining 1 tablespoon oil to the skillet. Add the mushrooms and cook, stirring occasionally, until they begin to brown and release their liquid, about 5 minutes. Add the garlic and cook, stirring, until the garlic is fragrant, about 30 seconds.
Add the cream cheese and stir until melted. Then pour in the half-and-half. Use a wooden spoon to scrape any browned bits from the bottom of the skillet. Season with salt and pepper. Add the chicken back to the skillet, cover and cook, turning about halfway through, until an instant-read thermometer inserted into the thickest part of the chicken reaches 165 degrees F, 15 to 20 minutes. Transfer the chicken to a serving platter filled with warm egg noodles. Add the spinach to the skillet, cover and cook until wilted, about 5 minutes. Stir to combine and season with salt and pepper. Pour the creamy spinach sauce over the chicken and noodles.
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