Heat a grill to medium high (about 450°) with an area cleared of coals (for charcoal grill) or a burner turned off (for gas) to make an indirect-heat area.
Trim pepper stems, leaving bases (calyxes) intact, then cut peppers in half lengthwise. With a small knife, carefully remove seeds and veins, keeping peppers' shapes intact to hold eggs. Brush insides with 1 tbsp. oil (total) and sprinkle with 1/2 tsp. salt.
Grill pepper halves, cut side down, over direct heat, covered, until grill marks appear and peppers are slightly softened, about 5 minutes. Transfer to a rimmed baking sheet while you grill bread.
Brush both sides of bread slices with 1 tbsp. oil (total). Grill over direct heat until lightly charred, about 1 minute per side. Transfer to a plate.
Arrange one or two large sheets of slightly crumpled aluminum foil on indirect heat. Nestle peppers, cut sides up, into foil to keep them steady. Divide cheeses and olives among peppers. Crack an egg directly into each pepper, then sprinkle with half of parsley and chives and remaining 1/2 tsp. salt.
Grill, covered, until white is set but yolk is still runny, 15 to 20 minutes. Top with remaining herbs, drizzle with oil, and serve with toasts.