Make the topping: Heat the oven to 375 degrees F. Put the breadcrumbs in a pie plate or on a small rimmed baking sheet, drizzle 1/2 tablespoon of the olive oil on top, and mix well to evenly coat the crumbs. Spread the crumbs and toast in the oven, stirring occasionally, until very crisp and golden brown, about 12 minutes. Let cool.

Combine the shallot, vinegar, and a pinch of salt in a small bowl. Let sit for at least 10 minutes and up to 2 hours.

Combine the remaining 1/4 cup oil with the herbs, capers, and anchovy in a medium bowl. Set aside until ready to serve.

Grill the eggplant: Prepare a medium-high charcoal or gas grill fire. Brush both sides of the eggplant slices with olive oil and season with salt. Grill (covered on a gas grill; uncovered on a charcoal grill) until golden-brown grill marks form, 3 to 4 minutes. Turn the eggplant and grill until tender and well marked on the second sides, 3 to 4 minutes more. The interior should be grayish and soft rather than white and hard.

Just before serving, combine the shallot mixture and the toasted breadcrumbs with the herb mixture. If the salsa seems too dry, add a bit more olive oil. Season to taste with more salt or vinegar, if necessary -- it should have a nice acidic kick. Spoon the salsa verde on top of grilled eggplant slices or serve on the side.

Make Ahead Tips: You can grill the eggplant in advance and serve it at room temperature, but wait until just before serving to combine the shallot mixture with the toasted breadcrumbs, so the breadcrumbs stay crunchy.