How to fry a potatoes with a mushrooms? Thank you.

I would fry them separately. I'd first par-boil tiny potatoes then smash and fry them. For the mushrooms, I'd cut in quarters then sauté in butter with garlic or shallots. I'd pass them separately potatoes first then mushrooms so each guest can pile the mushrooms over the potatoes. That way the mushrooms won't discolor the potatoes and each vegetable retains its integrity while enhancing the other.

Fry the mushrooms first, to cook out the liquid that mushrooms will ooze out. Then add the potatoes next. Add more oil if needed.

If you fry both at the same time, the liquid from the mushrooms will steam the food rather than fry it.