Whisk miso paste, rice vinegar, brown sugar, and soy sauce together in a bowl until glaze is smooth.
Place barramundi fillets, rounded-side up, onto a plate; spread 1/2 of the glaze onto rounded side of fish.
Heat oil over high heat in a nonstick skillet. Place fish, glazed-side down, in hot skillet and spread remaining glaze over other side; cook until slightly white around the edges, 1 to 2 minutes. Flip fish and cook until flaky and white, 1 to 2 minutes more.
