Drain the palm hearts and cut them lengthways into very thin strips, until the palm hearts look shredded. This doesn’t have to be perfect, but the strips will give you a fleshier texture.
Place the soaked cashews in the blender and add the water to cover them. Add the salt, pepper, Old Bay and cayenne and blend the cashews until smooth. Add the flour and blend again.
Transfer the mixture to a bowl, mix with the palm heart strips and form into 4 patties with your hands. Heat the oil in a frying pan over a medium-low heat. Add the patties and fry on each side until golden, 3 to 4 minutes per side (see Notes). Mix the tartar sauce ingredients and set aside.
To assemble, spread the tartar sauce on the buns, then add the lettuce, patties and cheddar-style cheese to make 4 sandwiches.
If you want the cheese to melt a bit, wrap each sandwich in foil for a few minutes; the cheese should melt on its own. Or, top the patties with the cheese as they finish cooking.