Whisk mayo and lemon juice together. Season with salt and pepper if desired.

Spread lemon-mayo over the inside of all 4 slices of bread. Divide the remaining ingredients equally and layer on 2 slices of bread, topping with the remaining 2 slices of bread.

Place assembled sandwich on an oiled/buttered panini press, close, and cook until the bread is toasted and cheese is melting (approx. 3-6 minutes depending on press).
(If you do not have a panini press you can also place this sandwich on a grill pan or saute pan over medium-low heat and cook each side while pressing with a heat-safe spatula or grill press for 3-4 minutes per side)