Bring 2″ water to a boil in a large saucepan; reduce heat to a gentle simmer and add vinegar. Crack an egg into a small bowl; gently slide egg into water. Repeat with remaining eggs, waiting until whites start to set before adding the next. Poach until whites are set and yolks are still runny, about 2 minutes. Using a slotted spoon, transfer eggs as they are done to paper towels. Heat 2 tablespoons butter in a large skillet over medium-high. Add half of both mushrooms; season with kosher salt and pepper. Cook, tossing occasionally, until mushrooms are tender and browned, about 5 minutes. Transfer to a plate. Repeat with remaining mushrooms and another 2 tablespoons butter; transfer to same plate. Drizzle with 1 tablespoon lemon juice. Heat remaining 2 tablespoons butter in same skillet over medium-high; cook sage, shaking skillet, until fragrant and crisp, about 2 minutes. Transfer to paper towels with a slotted spoon. Reserve skillet. Divide tamarillo slices among plates, arranging so they overlap; season with kosher salt and pepper. Drizzle with remaining 1 tablespoon lemon juice. If butter has started to solidify, return mushrooms to reserved skillet and gently reheat over medium, tossing. Using a slotted spoon, spoon mushrooms over tamarillos, top with poached eggs, fried sage, and Parmesan. Season with sea salt and more pepper. Drizzle any butter in skillet over eggs.