Arrange the slices of bread on a flat work surface and, using a small spoon, divide the pesto evenly among 1 side of each of the bread slices.
Divide the turkey, roasted pepper strips, and fontina equally among 4 of the bread slices. Top with the remaining 4 slices of bread, pesto sides down, to form 4 sandwiches.
Brush the outsides of each sandwich lightly with some of the olive oil.
Heat a large skillet, griddle, or grill pan over medium-high heat. When the skillet is hot, add the sandwiches and cook until the bread is golden brown and the cheese is melted, pressing occasionally with a large metal spatula or the bottom of a small heavy saucepan, about 4 minutes per side. Remove sandwiches and transfer to a cutting board. Slice in half on the diagonal and serve immediately.