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In a large skillet, heat the oil over medium-high heat. Sprinkle the chicken with the pepper and 2 teaspoons of the salt. Dredge in the flour, shaking off any excess. Working in batches as needed, place the chicken in the skillet and cook, turning once, until golden brown on both sides and an instant-read thermometer inserted in the thickest part of the meat registers 165°F, about 8 minutes per side. Transfer to a plate to cool and set aside.
Reduce the heat to medium and melt the butter in the same skillet. Add the shallots and garlic and sauté, stirring often, until just softened, about 2 minutes. Add the wine, scraping up any browned bits on the bottom of the skillet. Let simmer until the liquid is reduced by half, about 10 minutes. Add the cream and cook, stirring often, until the mixture can coat the back of a spoon, 5 to 8 minutes.
Remove the pan from the heat and add the spinach and remaining ¾ teaspoon salt. Stir until the spinach is wilted.
Cut the chicken against the grain into diagonal strips 1 inch wide, 2 to 3 inches long, and return it to the skillet. Sprinkle with the parsley and serve with the lemon wedges.
Store in an airtight container in the refrigerator for 3 to 5 days.