For each sandwich, lay 2 slices bread on a work surface. Lay about 3/4 ounce cheese on one slice of the bread; top with 2 ounces turkey meat, 2 tablespoons relish, about 5-6 arugula leaves and an additional 3/4 ounce cheese. Top with the other bread slice (there should be cheese next to each bread slice). Repeat with remaining ingredients.
Brush the top of the sandwich with olive oil; heat a non-stick skillet over medium heat. Put the sandwich, oiled-side down, into the pan, place a small heavy pan, or aluminum foil-wrapped brick on top of the sandwich to press it down. Cook until bread is golden and crisp and cheese is beginning to melt, about 5 minutes. Remove the heavy pan or brick, brush the top of the sandwich with oil and carefully flip it. Put the heavy pan or brick back on top of the sandwich and continue cooking until bottom bread slice is browned and crisp, cheese is melted, and sandwich is heated through, about 5 minutes more. Remove from the pan and cover to keep warm while cooking the remaining sandwiches.
To serve, slice each sandwich on the diagonal and serve warm.