Place lentils in a large saucepan and cover with cold water. Bring to a boil. Strain lentils and place back in pot.
Add carrots, onion, clove, bay leaf, broth, and a teaspoon of salt. Bring to a boil then reduce heat and simmer until lentils are tender, about 25 minutes. Drain pot. Place lentils and vegetables in a large bowl. Discard clove and bay leaf. Season to taste with pepper and more salt.
In a medium saute pan, heat olive oil over medium heat until shimmering. Add kale and cook, stirring frequently, until starting to tenderize, 3 to 4 minutes. Add garlic and cook, stirring constantly until fragrant, about 30 seconds. Season to taste with salt and pepper and add to lentils.
In a small bowl, whisk together olive oil, vinegar, and mustard. Season with salt and pepper. Pour vinaigrette over lentils and toss to combine. Serve immediately, or allow to cool and serve cold or at room temperature. Salad can be stored in a sealed container for up to 2 days in the refrigerator.