Add marinade ingredients to large ziplock bag (oil,oregano, sugar, lemon pepper seasoning, salt, pepper, garlic, and both the zest and juice from 1 lemon).
With a fork punch several holes in chicken breast so marinade can penetrate. Add chicken to bag and marinade for 4 hours.
Remove chicken from marinade and grill on charcoal/gas grill with a hunk of smoking wood until juices run clear. Remove and let chicken cool slightly to handle.
Put your palm flat on each chicken breast and slice each chicken breast in 1/2 so that each piece is 1/2 as thick as original chicken breast. Spread about 2 teaspoons of pesto over 4 slices of the bread. Add cheese and then 1/2 of each breast of chicken that you cut above. Add 3 artichoke per sandwich, broken up. Put other 4 slices of bread on top.
Grill in panini press until bread is nicely toasted and cheese is melted (about 4 minutes).