Stir together the artichokes, spinach, sour cream, fontina and Parmesan in a bowl. Season the mixture with salt and pepper to taste.
Preheat a grill pan or panini press on medium heat.
Spread the artichoke mixture on all the slices of bread. Place 1 layer of chicken on 4 of the bread slices, then top each with a couple of layers of roasted red peppers. Top with the remaining bread. Brush the outside of each sandwich with some olive oil.
Place the sandwiches on the grill pan or in the panini press. If using a grill pan, place a cast-iron skillet or brick wrapped in foil on top of the sandwiches to weigh them down. Cook, flipping halfway through, until golden brown and the cheese is melted and gooey in the center, 3 to 5 minutes per side. If using a panini press cook 6 to 10 minutes total.