Bring a large pot of water up to a boil. Cook the corn for 5 minutes, let cool, then use a knife to remove the kernels from each cob. Take one cup of corn and set aside.
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. Lower the heat, then add the garlic and rosemary and cook until fragrant, about 20 seconds. Add remaining corn, season with salt and pepper, then toss until coated in butter. Transfer to a blender, add 1/4 cup water, then blend on high until totally smooth. Taste for seasoning, adjust as needed, then set aside.
Use a vegetable peeler shave the zucchini into long ribbons. Add the remaining 2 tablespoons butter to the same saucepan over medium heat. Allow it to sizzle and brown, then add the zucchini, turn off the heat and gently toss to coat in the butter. Add the reserved 1 cup of corn and lemon juice, season with salt and pepper, then set aside while you cook the fish.
Heat vegetable 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, then season well with salt and pepper on both sides. When the oil is shimmering and almost smoking, gently place two fillets in the pan, (skin side down if using skin-on fillets). Use a spatula to gently press down on each fillet for about 15 seconds to prevent curling. Cook until opaque around the edges, then flip and finish cooking until opaque throughout.
Place a generous spoonful of corn puree on each plate, then top with the zucchini and corn mixture. Nestle the barramundi over top and garnish with a sprig of rosemary. Serve immediately.