Whisk miso paste, rice wine vinegar, brown sugar, and soy sauce together in a bowl until smooth.

Place barramundi fillets, rounded-sides up, on a plate; spread 1/2 miso glaze on rounded sides.

Heat oil over high heat in a nonstick skillet; add fillets, glazed-sides down, and spread remaining 1/2 glaze over flat sides fillets. Cook until slightly white around edges, 1 to 2 minutes. Flip fillets; cook until flaky and white, 1 to 2 minutes more.