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Step 1
Trim the stem ends of the leeks while keeping them intact. Cut off the dark green tops and remove any tough outer layers. Cut each leek in half lengthwise. Submerge in cold water for 5 minutes to remove any dirt from inside. Transfer to a colander and let drain.
Step 2
Melt the butter in a large frying pan over medium-low heat. Add the vermouth. Place leeks cut-side down in the pan and cook for 4 minutes. Add the thyme and broth. Bring to a gentle boil, then reduce to a simmer.
Step 3
Cover and cook until leeks are fork-tender, 10 to 15 minutes. Add the lentils and cook for 5 minutes more. Taste and season with salt and pepper.
Step 4
Remove and discard the thyme stems. Transfer the leeks with tongs to a serving platter. Pour the lentils and cooking liquid over the leeks and garnish with the parsley.