Heat the oil in a large Dutch oven or soup pot over medium-high heat. Once hot, add the onion, along with a pinch or two of salt. Cook until onion has softened and golden brown, 6 to 8 minutes.
Add the carrots and garlic and another pinch of salt. Cook until the carrots are just starting to soften, 2 to 3 minutes.
Pour in a bit of the vegetable broth to deglaze the pot and scrape up any browned bits. Add the remaining broth, lentils, potatoes, bouquet garni, crushed tomatoes (including their juices), 1 ½ teaspoons kosher salt, and pepper to taste. Stir well.
Bring the soup to a boil. Then reduce heat to low, cover the pot partially, and simmer the soup for 40 to 50 minutes, or until the lentils are tender.
Remove the bouquet garni. Stir in the dijon mustard and balsamic vinegar and taste. Add more salt and pepper to taste.