Place the squash in a large pot and cover with cold water. Bring up to a boil and simmer for about 20 minutes or until the squash is fork tender. Drain and set aside in a bowl.
Melt 6 tablespoons of butter in a medium saucepan and keep a close eye on it as it sizzles, foams, and eventually turns brown and smells nutty.
Pour the browned butter over the squash and season with salt and pepper. Use an immersion blender to puree until very smooth. Taste and adjust seasoning as needed, then set aside until ready to serve.
Add 1 teaspoon olive oil to a medium saucepan set over medium heat and saute the shallots and rosemary until tender, about 3 minutes. Add the white wine and simmer over medium high heat until almost very little liquid remains, about 5 minutes. Add the pomegranate juice and continue to reduce until there is only about 3/4 cup liquid, about 15 minutes. Strain out the shallots and rosemary, season with salt and pepper and return to the pot.
Heat 3 tablespoons olive oil in a large skillet (well-seasoned cast iron works best) over medium-high heat. Use paper towels to pat the barramundi fillets very dry, being sure to remove as much moisture as possible. Season both sides with salt and pepper. When the oil is shimmering and almost smoking, gently place two fillets in the pan, (skin side down if using skin-on fillets).
Allow to cook in the hot oil without moving for about 3 minutes.
The fish will be ready to flip when it easily releases from the pan. If it sticks, it’s not ready. Lower the heat and give it another minute. Use a fish spatula to gently flip the fillets and cook on the other side for another 2-3 minutes, depending on thickness. Transfer to a plate and repeat with remaining fillets.
Whisk the remaining 3 tablespoons of butter into the pomegranate sauce over low heat, then taste for seasoning and adjust as needed.
To serve, place a generous dollop of squash puree on each plate, then top with barramundi. Drizzle the pomegranate sauce around the perimeter and garnish with pomegranate seeds and rosemary sprigs if desired. Serve immediately.