Prepare leeks. Trim roots if overly long, but leave enough to keep bases intact. Slice off most of the green tops. Slice leeks in half lengthwise. Rinse under running water, gently fanning layers to wash out any trapped grit. Leeks like to grow in sandy soil, so you need to clean them carefully.
Heat a sauté or frying pan large enough to hold leeks in a single layer over a medium flame. Add oil and butter and swirl together. Arrange leeks in pan, cut side down, and sauté, turning occasionally with tongs and spatula, until lightly browned, about 5 minutes. Handle leeks gently when turning to keep as intact as possible.
Add stock, water and 1 tablespoon parsley to pan. Bring to a simmer, reduce heat and cover pan, cooking leeks until tender, about 10 minutes. Arrange leeks on a serving platter, cut side up.
Make the vinaigrette, adding lemon juice and mustard to pan, whisking to combine. Season vinaigrette with salt and pepper [use a light hand with the salt and taste before seasoning—your stock will provide some salt]. Pour vinaigrette over leeks and sprinkle with remaining parsley. Serve immediately.