Make about 8 slices lengthwise down the apple halves but not all the way through one end, so that they are still in one piece and will fan out when softened by cooking.
In a large bowl, combine the sherry vinegar, shallot, and salt and let sit for 5 minutes to let the flavors mellow. Add the pepper and whisk in the olive oil and herbs. Add the apples to the bowl and toss to coat them with the vinaigrette. Cover the bowl with plastic wrap and let the apples sit at room temperature for at least 1 hour and up to 2 hours.
Prepare a grill or heat a grill pan over medium-high heat. Remove the apples from the vinaigrette and reserve the vinaigrette. Lay the apples on the grill, flat side down, and grill them until browned, 4 or 5 minutes. Turn the apples and brown the other side for another 5 minutes. The apples should be caramelized and softened.
There should be about 8 tablespoons of vinaigrette left in the bowl. Remove 3 tablespoons of the vinaigrette and combine it with the olives in a food processor, mini chopper, or blender. Process until you have a fine paste. Thin the paste with water, if desired. Transfer the olive dressing to a squeeze bottle with a large opening, or reserve it in the work bowl.
In a medium bowl, toss the remaining vinaigrette with the micro greens. Arrange the greens on the center of 6 plates and top each with an apple half, slightly fanned out, cored side up. Drizzle some of the olive dressing over the top, or squirt it decoratively with the squeeze bottle. Serve immediately.