Spread Rémoulade on 4 slices of bread. Place 3 slices ham on 2 slices of bread.
In a bowl, add sliced avocado and squeeze lime juice. Toss to coat before placing the slices on top of the ham, covering the entire surface.
Place red onion slices, hard-boiled egg slices, tomato slice, cheese slice, and butter lettuce. Place the other slices of bread on top, pressing down to compact and butter the top ones covering the entire surface.
Place the sandwiches in the panini pan, buttered side down, put the lid on and grill about 4 minutes or until nice and brown.
Spread the butter evenly, carefully flip the sandwiches, put the lid back on and grill until golden. Serve them immediately.