Preheat the oven to 350ºF. Spray 12 muffin tins with cooking spray (do not use paper liners).
Rinse and strain the lentils. Place them in a pot with 3 cups of vegetable broth or water. Bring to a boil and season with salt. Reduce the heat to medium low and simmer, uncovered for at least 40 minutes or until the water is absorbed and the lentils are slightly mushy. Stir frequently and add a bit of water if they begin to dry out before 40 minutes. Mash the lentils slightly with a spoon when ready.
While the lentils are cooking, combine breadcrumbs and milk in a large bowl and let soak. Heat 2 tablespoons olive oil in a large skillet over medium heat. When hot, add the onion and cook, stirring occasionally, until soft but not brown, 5 to 7 minutes. Add the celery and cook 3 to 4 minutes longer. Add the mixture to the breadcrumbs.
In a small pan, toast walnuts over medium-low heat, shaking the pan occasionally, until fragrant, about 5 minutes. Add the nuts to the breadcrumb mixture along with the carrot, Parmesan, 1/4 cup ketchup, thyme, oregano, salt, pepper, and lentils. Mix thoroughly and divide evenly amongst muffin tins.
In a small bowl, combine 1/4 cup ketchup and 2 tablespoons balsamic vinegar. Spoon mixture over the loaves.
Bake until the loaves are lightly browned, about 30 minutes.