Heat 3 tablespoons of the olive oil in a large skillet over medium heat. Add the onions, celery and carrots and season with salt and pepper. Cook until the vegetables have softened, 5 to 6 minutes. Add the garlic and cook for a minute. Add the brisket, collards, chicken broth, tomatoes and sugar. Simmer, covered, over low heat for 12 minutes. Taste and adjust the seasoning as needed.
In two batches, heat half (3 tablespoons) of the remaining olive oil in a large skillet over medium-high heat. Add half of the pasta, lower the heat to medium and cook until lightly browned and crispy at the edges, 5 to 6 minutes per batch.
To serve: Toss together the brisket ragu and the crispy pasta, and divide it between four plates. Garnish each serving with a basil leaf and two pieces of shaved Parmesan.