In a large Dutch oven, heat 1 tablespoon extra-virgin olive oil over medium heat. Add the carrots, celery and onion. Sauté until they start to soften a bit, about 5 minutes.
Add the sausage and continue to cook, breaking up the sausage as it cooks, until the sausage is nicely browned. Season with salt and pepper.
Add the lentils and chicken broth. Bring to a simmer and cook according to the instructions on your lentil package. You want to cook your lentils until they are still al dente, just a bit of give, but not mushy. This should take about 15-20 minutes. Taste as you go, and when the lentils are cooked it's time to add the rest of the ingredients.
Add the coconut milk and bring to a simmer. Once it thickens a bit, stir in the kale, add lime juice, and let it all simmer for about 5-10 minutes, until the kale is wilted.
Serve immediately.