Place potatoes in a medium saucepan and add cold water to cover by 1″. Bring to a boil, season with salt, and cook until fork-tender, 15–20 minutes. Transfer potatoes to a plate with a slotted spoon.
Return water to a boil and cook haricots verts in same saucepan until crisp-tender, about 2 minutes. Using a slotted spoon, transfer to a bowl of ice water. Chill until cold, about 3 minutes. Transfer to paper towels and pat dry.
Return water in pot to a boil and cook eggs 8 minutes. Transfer eggs to bowl of ice water and chill until cold, about 5 minutes. Peel; set aside.
Meanwhile, heat 2 tablespoons oil in a small saucepan over medium-high. Add capers and cook, swirling pan occasionally, until capers burst and are crisp, about 5 minutes. Transfer capers with a slotted spoon to paper towels to drain. Let oil cool; set aside.