Combine the lentils, 5 cups broth, bay leaf, and 1 teaspoon of the salt in a saucepan. Bring to a boil, reduce the heat to medium-low, and cook until the lentils are just tender, about 20 minutes.
While the lentils are cooking, pour the olive oil into a large deep skillet over medium-high heat. When it shimmers, add the onion, garlic, and carrot and sauté until the vegetables are tender, about 5 minutes. Stir in the remaining 1 teaspoon of the salt and the mustard seeds, then stir in the mustard greens and cook until they fully wilt, about 5 minutes.
When the lentils are tender, drain them, discard the bay leaf, and add them to the mustard greens mixture in the skillet, stirring to combine. Turn off the heat and stir in the shallot, vinegar, and Dijon mustard. Taste and add more salt if needed.
To serve, divide the salad among six serving plates and top with the pistachios and feta.