Split rolls and set aside.
Combine giardinera and olives in food processor and blend until chunky (too much blending will give you soup). Spoon about 1 Tbs. onto roll and then proceed with one slice cheese and then one slice each of the meat. End with another slice of cheese and then add the roasted peppers and artichokes.
Flatten and place on panini press or use a cast iron skillet that has been heated sprayed with cooking spray. Add sandwiches (two at a time) and place a brick or another skillet on top. If using skillet, turn after about 1 minute and cook an additional 1-2 minutes on second side. Heat until golden and cheese is melting.