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 green beans 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 quail eggs, 4 minutes. Transfer eggs to bowl of ice water and chill until cold, about 5 minutes. Peel; set aside.
Meanwhile, heat 2 Tbsp. 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.
Meanwhile, mix mustard, and sugar in a large bowl to form a coarse paste. Remove peel and white pith from lemon. Working over bowl, cut between membranes of lemon to release segments. Squeeze remaining juice from membranes into bowl; discard membranes. Whisk in shallot, then slowly whisk in remaining ¼ cup oil. Season vinaigrette with salt and pepper.
Halve reserved potatoes and eggs crosswise.
Arrange on a platter; season with salt and pepper. Drizzle with half of dressing. Top in separate piles with potatoes, green beans, hard-boiled eggs, olives, and candied salmon. Drizzle salad with remaining dressing and top with fried capers.