Preheat oven to 450°F (232°C) with rack set in lowest position.
Add ½ pound fingerling potatoes to large sauté pan, cover with water (they should be completely submerged) and bring to a low boil over high heat.
Meanwhile, finely dice 1 small shallot. Transfer to a small bowl and cover with cold water; set aside.
Once potatoes are boiling, cook until a toothpick easily slides in and out of potatoes, about 15 minutes. Using a spider or slotted spoon, transfer potatoes to a strainer to dry.
Add 8 ounces green beans to simmering water and cook 4 minutes. Meanwhile, prepare an ice water bath.
Using a slotted spoon or spider, transfer green beans to water bath, submerge in water and set aside.
Place potatoes on a baking sheet and using bottom of a cup, smash potatoes until they are ¼-inch thick. Drizzle potatoes with 2 tablespoons avocado oil, shimmy potatoes to coat in oil then flip and shimmy again. Season both sides of potatoes with heaping ½ teaspoon kosher salt. Season tops with ¼ teaspoon pepper. Bake until bottoms start to brown, about 15 minutes. Flip and continue to cook until browned and crispy around edges; 10–15 minutes more.
Drain shallot and add to a medium mixing bowl with drained and dried blanched green beans, ½ cup torn olives, 1 tablespoon olive oil, 1 tablespoon chopped fresh dill, minced zest of 1 lemon, and 1 tablespoon lemon juice; season to taste with salt and pepper.
Spread ½ cup whole milk Greek yogurt over a large serving plate.
Layer smashed potatoes over yogurt then arrange green bean mixture over top. Serve immediately.