Heat 1 tablespoon olive oil in a large nonstick skillet; cook and stir chicken breast and onion until heated through, 3 to 4 minutes. Transfer to a plate.
Wipe out skillet with a paper towel. Heat 1 tablespoon olive oil until bubbling. Add 2 bread slices; sprinkle half of the fontina cheese on top. Cover cheese with half of the chicken and onion. Cook until bread is toasted and cheese is melted, 1 to 2 minutes.
Layer pear slices over chicken and onion; press bread slices together to make a sandwich. Repeat with remaining oil, bread slices, fontina cheese, chicken, onion, and pear slices to make second sandwich.