In a shallow dish, whisk together the flour, salt, and pepper. Pat the chicken breasts dry and dredge each one in the flour mixture, shaking off any excess.
Heat the olive oil in a large skillet over medium-high heat. Add the chicken and cook until golden brown, about 4-5 minutes per side. Transfer the chicken to a plate.
Reduce the heat to medium. Add the mushrooms and sliced red pepper to the same skillet. Cook, stirring occasionally, until they have softened, about 5-6 minutes.
Add the minced garlic and cook for 30 seconds until fragrant.
If using, pour in the white wine to deglaze, scraping up any browned bits from the bottom of the pan. Let it bubble for about a minute.
Stir in the chicken broth and lemon juice. Bring the sauce to a simmer.
Return the chicken to the skillet. Add the drained artichoke hearts and lemon zest. Reduce the heat to low, cover, and simmer for 10-15 minutes, or until the chicken is cooked through.
Confirm the chicken has reached an internal temperature of 165°F (74°C) with a meat thermometer. Remove the skillet from the heat.
Stir in the fresh tarragon and serve immediately with the pan sauce spooned over the chicken.