1/3 cup (packed) fresh basil leaves
1/2 small garlic clove
6 tablespoons extra-virgin olive oil
4 large eggs
2 thick slices brioche, toasted
Parmesan cheese shavings
Puree basil, garlic, and oil in mini processor until very smooth. Season pistou to taste with salt and pepper.
Add enough water to medium pan to measure roughly 3 cm. 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 slotted spoon, transfer 2 eggs, well drained, to each piece of brioche. Sprinkle eggs with salt and pepper. Drizzle with some of pistou and serve.