Sprinkle chicken evenly with 1 teaspoon of the salt and ½ teaspoon of the pepper. Heat 2 tablespoons of the oil in a large Dutch oven or heavy-bottomed pot over medium-high. Working in 2 to 3 batches, add chicken and cook, turning occasionally, until lightly golden but not yet cooked through, 4 to 6 minutes total. Transfer chicken to a plate; repeat with remaining chicken, adding remaining 1 tablespoon oil to pot if needed to keep meat from sticking. Set cooked chicken aside.

Reduce heat to medium and add shallots. Cook, stirring often, until softened, about 3 minutes. Add garlic, capers, and crushed red pepper; cook, stirring constantly, until fragrant, about 30 seconds. Sprinkle shallot mixture with flour; cook, stirring constantly, until evenly coated, about 30 seconds. Add wine; cook, scraping any browned bits from bottom of pot, until thickened, about 30 seconds to 1 minute. Gradually pour in broth, stirring constantly. Stir in artichokes, spinach, reserved chicken, remaining 1 teaspoon salt, and remaining ½ teaspoon pepper; bring to a boil over medium, stirring occasionally. Once simmering, cover and reduce heat to medium-low to maintain a gentle simmer. Cook, covered, and stirring occasionally, until chicken is cooked through and tender and flavors have melded, 15 to 20 minutes.

Uncover, remove from heat, and gradually stir in boursin cheese, Parmigiano-Reggiano, mustard, and ½ cup of the cream until melted and creamy. If desired, gradually stir in up to remaining ¼ cup cream until desired consistency is reached; season to taste with salt. Garnish with additional Parmigiano-Reggiano and crushed red pepper; serve hot with crusty bread.