Salmon Cakes with Lemon-Caper Yogurt Sauce

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Salmon: 

1 pound skinless salmon fillet (see Cook's Note)

Olive oil, for drizzling

Kosher salt and freshly ground black pepper

Kosher salt and freshly ground black pepper 

Patties:

1 large egg, beaten

1/3 cup (1/2-ounce) chopped fresh chives

26 saltine crackers, crushed, divided

1/2 cup frozen corn, thawed

2 tablespoons Dijon mustard

3 tablespoons mayonnaise, plus more, as needed

1 tablespoon capers, rinsed and drained

1 tablespoon lemon juice

1 tablespoon lemon zest

3 tablespoons vegetable oil

3 tablespoons unsalted butter, at room temperature

Sauce:

1/2 cup full-fat plain Greek yogurt

1 1/2 tablespoons capers, rinsed, drained and chopped

1 tablespoon lemon juice 

1 tablespoon lemon zest

Kosher salt and freshly ground black pepper