Preheat the oven to 350 degrees.
Pour white wine into a large sauté pan with high sides and place over medium-low heat. Add the butter, garlic, and thyme, then swirl the pan around and allow the butter to slowly melt.
Add the vegetables, then gently toss to coat with butter. Poach for about 5 minutes, then use a slotted spoon to transfer the vegetables to a roasting pan and finish cooking in the oven (leave the garlic and thyme behind). Cook-time will vary depending on the vegetables and their size. Check after 10 minutes - they’ll be done when a fork can easily pierce the center.
Carefully place the fish fillets in the butter. Season with salt and pepper, then swirl the pan around so butter coats the top of each fillet. Adjust the heat to keep the butter just below a simmer in order to prevent it from breaking. Allow the fish to poach slowly until it becomes opaque. Thicker fillets may be flipped over to finish cooking on the other side.
Spoon vegetables on to a plate, then carefully place a barramundi fillet over top. Drizzle some of the butter sauce around the plate, then sprinkle the fish with flaky sea salt and parsley. Serve with lemon wedges on the side if desired.