Preheat the oven to 350F.
Put the leek rounds in a large bowl of tepid water and swish so that any dirt falls to the bottom of the bowl. Set a medium saute pan over medium-high heat, lift the leeks from the water, drain, and add them to the pan. Season with salt and cook, stirring often, for about 5 minutes. As the leeks begin to soften, lower the heat to medium-low. The leeks will release liquid. Stir in the butter to emulsify, and season with pepper to taste. Cover the pan with a parchment lid, and cook, stirring every 10 minutes, until the leeks are very soft, 30 to 35 minutes. If at any point the butter breaks or looks oily, stir in about a tablespoon of water to re-emulsify the sauce. Remove and discard the parchment lid.