Heat oil in a large Dutch oven over medium heat. Add sausage; using a spatula, press the sausage into an even layer in the pot. Cook, undisturbed, until golden brown on the bottom, about 4 minutes. Using a spoon or spatula, stir the sausage to crumble it into smaller pieces; cook, stirring occasionally, until no longer pink, about 3 minutes. Add leek, celery and carrots; cook, stirring often, until the vegetables are tender, about 10 minutes. Stir in garlic, thyme, pepper and salt; cook, stirring constantly, until fragrant, about 1 minute. Add tomato paste; cook, stirring constantly, until the tomato paste is darkened and caramelized, about 2 minutes.
Stir in broth, collards, lentils and tomatoes; bring to a boil over medium-high heat. Reduce heat to medium-low and simmer, covered, until the flavors meld and the collards are tender, about 20 minutes.
Remove from heat; stir in vinegar. Divide the soup among 6 bowls; sprinkle with Parmesan. Garnish with the reserved celery leaves, if desired.
