Preheat the oven to 400°F. Line a baking sheet with parchment paper and set aside.
Place the beets in a large pot and cover them with water. Bring the beets to a boil. Lower the heat to a simmer and cook until very tender, about 30 minutes. Take the beets out of the water with tongs and let them cool a bit on a cutting board.
Once cool enough to handle, slip the skins off of the beets. Then, transfer the beets to the parchment-lined baking sheet. Using the bottom of a wide bowl or jar, gently crush the beets until they break off into craggy fragments. The pieces will be all different shapes and sizes and this is okay! Spread the pieces out in an even layer on the baking sheet.
Drizzle the crushed beets with 2 tablespoons of the olive oil. Season the beets with salt and pepper. Slide the baking sheet into the oven for 20 minutes, or until the beets have dried out a tiny bit on the surface.
While beets are roasting, make the dressing: In a medium bowl, combine the olives, shallot, capers, lemon zest, lemon juice, remaining ¼ cup of olive oil, and some ground black pepper. Use a fork to “whisk” and mix it up. Set aside.
Transfer the roasted and crushed beets to your serving platter, placing them in one layer.
Peel, pit, and dice the avocado, and the scatter the diced pieces evenly over the beets. Quickly stir the mint and parsley leaves into the dressing, and then spoon the dressing over the crushed beets and avocado. Serve immediately!