Season chicken with salt and pepper. Melt 1 tablespoon butter in Dutch oven over medium-high heat. Brown chicken on all sides, about 6 minutes; transfer to bowl.
Melt remaining 2 tablespoons butter in now-empty pot over medium heat. Add leeks and couscous and cook until leeks have softened and couscous is lightly toasted, 4 to 6 minutes. Stir in broth and 1 teaspoon salt and bring to simmer. Reduce heat to medium-low, cover, and cook, stirring occasionally, until couscous is tender, about 12 minutes.
Stir in chicken and any accumulated juices and cook, covered, until chicken is cooked through, about 5 minutes. Off heat, stir in spinach, Parmesan, peas, and cream and let sit until heated through, about 5 minutes. Serve.
