(Makes enough for 2 hearty side dishes)
Extra virgin olive oil, about a tablespoon
1/3 cup or so white wine
First, prep the vegetables. Rinse the leeks well, then slice the white and light green part of the stalk into rounds, then halve the rounds. Save the dark green ends for stock (and if you're roasting a chicken, you should definitely be making stock). Fill a large bowl with cold water and soak the chopped leeks. Leeks tend to have quite a bit of grit in them, and soaking helps get rid of that.
While the leeks soak, heat a skillet on medium high and add olive oil to coat the bottom of the pan. While the skillet heats, cut carrots into half moon rounds (might as well save the carrot tops for that stock too). When the oil in the pan has heated, add drained leeks and carrots. Cook, stirring occasionally, until starting to soften and brown at the edges, about 7 minutes. Add white wine and turn heat to low. Cook, stirring occasionally, until most of the wine has cooked off and carrots are tender.
Add salt and pepper to taste and serve.