Build the Aromatic Foundation: In a large Dutch oven or heavy-bottomed pot, warm the ⅓ cup of olive oil over medium heat. Add the sliced garlic, diced carrot, and diced celery. Cook, stirring often, for 6-8 minutes, until the vegetables have softened and the garlic is fragrant and lightly golden.
Bloom the Umami: Add the rosemary sprig and red pepper flakes, stirring constantly for 30 seconds until aromatic. Stir in the tomato paste and cook for 2 minutes, allowing it to toast and deepen to a dark, brick-red color.
Deglaze and Emulsify: Pour in the white wine, using a wooden spoon to scrape any flavorful browned bits from the bottom of the pot. Allow the wine to simmer and reduce by about half, about 2 minutes, creating a thick, emulsified base.
Create the Chickpea Velouté: Remove the pot from the heat and discard the rosemary sprig. Add one-and-a-half cups of the chickpeas (the equivalent of one can) and 2 cups of the unsalted stock. Using an immersion blender, carefully blend the mixture directly in the pot until it is completely smooth and creamy.
Simmer the Flavors: Return the pot to medium-high heat. Stir in the remaining chickpeas and the remaining 4 cups of stock. Season generously with freshly ground black pepper and bring the mixture to a lively simmer.
Cook the Pasta: Add the whole wheat ditalini to the simmering pot. Cook, stirring frequently to prevent sticking, for about 8-10 minutes, or until the pasta is al dente and the liquid has thickened into a rich, stewy sauce. If the mixture becomes too thick, add a small splash of water or more stock to reach your desired consistency.
Finish and Serve: Remove the pot from the heat. Stir in the grated Parmigiano-Reggiano and the fresh lemon juice until the cheese is fully melted. Ladle into warm bowls, drizzle generously with extra-virgin olive oil, and serve immediately.