Spread bottom half of focaccia with olive paste. Arrange chicken on top of paste. Arrange artichokes, tomatoes, peppers, and prosciutto over chicken. Sprinkle with cheese. Top with top half of focaccia; press gently.
Heat a large nonstick skillet over medium heat. Add sandwich to pan. Place a cast-iron or heavy skillet on top of sandwich; press gently to flatten. Cook 2 minutes on each side or until bread is lightly toasted (leave cast-iron skillet on sandwich while cooking). Cut into 4 wedges.