Boil the potatoes in a large pot of salted water. Remember to put the potatoes in cold water, let them come to a boil, and cook until fork tender. Drain the potatoes and let them cool down, then peel them (or not if they are new potatoes).
Rinse the flounder under cold water and pat dry with a paper towel.
Coat each flounder filet in flour, being sure to shake off any excess flour, or it will burn in the pan and make a mess.
Melt five tablespoons of the butter in a large skillet or fry pan big enough to hold all four flounder filets. Use two pans or cook in batches if your pan is not big enough. Pan-fry the fish for about 5 minutes on each side or until the filets are golden brown. If cooking in batches, keep the cooked filets warm while you finish the rest. Here you can use another pan to finish the potatoes or remove the fish to a plate and keep it warm. Add the remaining four tablespoons of butter to the pan, add the potatoes and sauté a minute or two. Next, add the minced fresh parsley and stir to coat.
Season with salt and pepper, to taste, and serve immediately with the flounder topped with slices of lemon and dill sprigs.
This is so simple to prepare and extremely delicious especially if the potatoes and fish are ultra-fresh.