Begin by washing the baby red potatoes thoroughly under cold water to remove any dirt. Cut them into 1" pieces, quartering any larger potatoes to ensure even cooking.
In a large pot, bring water to a rolling boil and add 1 teaspoon of salt. Carefully add the cut potatoes to the boiling water. Allow them to cook for about 10-15 minutes, or until they are fork-tender but not mushy.
Once the potatoes are cooked, drain them in a colander and rinse briefly under cold water to stop the cooking process. This will also help to keep the potatoes firm.
In a large skillet, melt 4 tablespoons of unsalted butter over medium heat. Once the butter is melted and bubbling, add the drained potatoes to the skillet. Toss to coat the potatoes evenly in the butter.
Sprinkle garlic salt over the potatoes to taste. Start with a small amount and adjust according to your preference. Continue to sauté the potatoes for about 5 minutes, allowing them to get a bit crispy and golden.
Once the potatoes are golden brown, remove them from the heat. Stir in the chopped fresh dill and shredded Parmesan cheese. The warmth of the potatoes will help melt the cheese and infuse the dish with a rich flavor.
Transfer the Buttered Red Potatoes with Dill to a serving dish. Garnish with a little extra dill on top for presentation, and serve immediately while hot and buttery.