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 5 tablespoons of the butter in a large skillet or fry pan big enough to hold all 4 flounder filets. If your pan is not big enough, use two pans or cook in batches. Pan-fry the fish for about 5 minutes on each side or until the filets are golden brown. If cooking in batches, be sure to 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 warm. Add the remaining 4 tablespoons of butter to the pan, add the potatoes and sauté a minute or two. Add the minced fresh parley 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.