Let's cook.
Trim the dark green tops from the leeks and slice the remaining white and light green parts in half lengthwise. Rinse well under cold water to remove any grit.
Melt the butter in a large skillet over medium heat. Place the leeks in the pan cut-side down and cook for 5 minutes until lightly browned.
Pour in the vegetable broth and sprinkle with fresh thyme. Reduce heat to low, cover, and braise for 12-15 minutes until the leeks are very tender.
Remove the lid and increase the heat slightly to reduce the remaining liquid by half.
Whisk the heavy cream and Dijon mustard together in a small bowl, then pour the mixture into the skillet around the leeks.
Simmer for 2-3 minutes until the sauce thickens slightly and coats the leeks. Sprinkle with Parmesan cheese before serving.
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