Spread 1 side of each bread slice evenly with mayonnaise. Place 1 bread slice, mayonnaise side down, on a sheet of parchment paper or wax paper. Top with shredded fontina and goat cheese. Cover with remaining bread slice, mayonnaise side up. Place a second sheet of parchment paper or wax paper on top of sandwich, and press gently.
Heat a large cast-iron skillet over medium. Remove sandwich from bottom piece of parchment paper, leaving top piece in place, and transfer to hot skillet. Place a small, heavy skillet on top of sandwich, and press. Cook until cheese melts and bread is golden brown on bottom, about 2 minutes. Remove small skillet; turn sandwich, and cook, without pressing, until golden brown on bottom, about 2 minutes.