What purpose does an egg play in a frangipane and what can I replace it with?

I'm cooking for a medical diet that excludes eggs, nuts, seeds, dairy, and other things. Coconut is ok, and I've seen a few recipes replacing almond flour with coconut flour and butter with coconut oil.

What role does the egg play and what can I replace it with?

Things that can't be used: chia seeds, flax seeds, gelatin (because the right kind can't be sourced at the moment).

What's left? Could I use apple sauce or the apple pulp from a juicer?