Puree basil, garlic and oil in a mini processor until very smooth. Season pesto to taste with salt and pepper.
Add enough water to a medium skillet to measure 1-¼ inches. Sprinkle salt generously into water. Bring water to simmer over medium heat. Crack eggs one at a time and gently slip into water. Cook until egg whites are just set and egg yolks are still runny, about 3 minutes.
Place hot toasts on plates. Top each with Parmesan. Using a slotted spoon, transfer 2 eggs, well drained, to each piece of brioche. Sprinkle eggs with salt and pepper. Drizzle with some of the pesto and serve.