Trim the leeks of their root and green tops, then slice lengthwise. If you see any grit, wash the leeks and shake them dry. Cut them crosswise into very thin half-ring slices.
In a braiser or heavy saucepan, melt a good knob of butter in olive oil over gentle heat, then add the leeks, along with a generous pinch of salt and a good grind of black pepper. Give the leeks a good mix and let them sweat for a minute or two. Then add a drizzle of water and cover. Let the leeks stew for 10-15 minutes, until they are very soft, stirring from time to time and adding another drizzle of water if things get too dry. Take care not to brown the leeks at all.
When the leeks are soft, uncover and turn the heat up a bit to let any excess moisture cook off. When you just start hearing the pot sizzle, turn off the heat.