Heat the oil in a large stock pot (at least 8 quarts) over medium-high heat until shimmering.
Add onions, carrots, celery, and 1 tsp. salt. Reduce heat to medium-low, cover, and cook for 20 minutes, stirring occasionally.
Stir in garlic until fragrant, about 30 seconds. Add water, broccoli, cauliflower, turnips, basil, bay leaves, and peppercorns. Bring to a gentle simmer and cook uncovered for 1 hour, stirring occasionally.
Strain the solids from the broth in batches using a fine-mesh sieve. Stir in 2 tsp. salt or season to taste.
Proceed with a recipe or cool completely and refrigerate for up to 3 days. The broth can be frozen for 1 month. Thaw before reheating.