Preheat the oven to 475°F with a rack in the center position. Grease a rimmed sheet pan with olive oil.
Add a generous pinch of salt to a large pot of water and bring to a boil over high heat. Add the potatoes and cook until just tender, 15 to 20 minutes. Strain the potatoes in a colander and let cool for about 5 minutes.
Meanwhile, make the gribiche. Using the side of a knife, smash the cucumber pieces until they’re roughly broken. In a medium bowl, stir together 3 tablespoons of the olive oil, the vinegar, mustard, and 1 teaspoon of salt. Add the smashed cucumbers, the eggs, olives, and parsley and toss to combine.
Spread out the potatoes on the prepared sheet pan. Using the bottom of a drinking glass, smash the potatoes until each is about ¼ inch thick. Drizzle the remaining 3 tablespoons oil all over the potatoes and sprinkle them with 1 teaspoon salt.
Bake the potatoes for 15 minutes, then add the asparagus to the sheet pan (overlapping is okay) and bake for about 10 more minutes, or until the potatoes are crispy and the asparagus is tender.
To serve, transfer the potatoes and asparagus to a serving platter. Top evenly with the gribiche and freshly cracked black pepper.