Heat an inch or so of water to low boil. Salt water and boil asparagus 2 minutes. Cold shock and drain.
Whisk together eggs, lemon juice and cheese. Season with salt and pepper.
Form sandwiches stacking 1 slice of bread, 1 slice cheese, 1 slice ham, asparagus, 1 slice ham, 1 slice cheese, another slice of bread.
Heat a thin layer of oil, a turn of the pan, over medium heat. Melt a tablespoon of butter into oil.
Coat 2 sandwiches in egg batter, like you are preparing a batch of French toast. Add to skillet and cook 5-6 minutes on each side until deeply golden and the cheese has melted. Repeat with remaining sandwiches.
Combine warm honey with mustard and drizzle over halved sandwiches or serve warm honey and mustard as dipper in ramekins. Serve with an arugula and cherry tomato salad alongside.