Preheat oven to 450 degrees. Melt butter in a large skillet over high. Add leeks; cook, stirring, until just golden, about 4 minutes. Add chard; season with salt and pepper. Cook, tossing, until wilted, about 3 minutes. Let cool.
In a small bowl, whisk together yogurt, lemon juice, mustards, garlic, and soy sauce.
Butterfly salmon by cutting horizontally through middle along one long side, leaving other side intact. Open flat, like a book. Flip salmon, skinned-side up; season with salt and pepper. Spread top with 1/4 cup yogurt mixture. Top with dill sprigs, then chard mixture.
With a long side facing you, roll up salmon into a tight log. Secure with 6 pieces of kitchen twine, starting about 1 1/4 inches from end and spacing evenly. Cut salmon into 6 equal pieces, slicing between twine. Tuck a dill sprig under twine on each roll.
Place rolls in a baking dish and drizzle with oil, rubbing to coat. Season with salt and pepper. Roast until salmon is opaque on outside but still pink inside, about 15 minutes. Serve, with remaining yogurt mixture alongside.