Preheat the oven to 300 degrees F (150 degrees C).
Allow the Unsalted Butter (1/4 cup) to soften, and wash and remove the ends of the Asparagus (1 bunch).
Gently separate the Garlic (1 bulb), but do not peel the cloves. Place them on aluminum foil, drizzle with 1 tbsp of Extra-Virgin Olive Oil (3 tablespoon) and season with Kosher Salt (to taste) and Ground Black Pepper (to taste).
Wrap the garlic up in the aluminum foil, and cook for 50 – 60 minutes. When the garlic is very soft, remove from the oven. Once the garlic is cool enough to handle, squeeze the cloves out of the skins and mash.
In a small bowl, combine the unsalted butter, White Miso Paste (1/4 cup) and roasted garlic. Mix until smooth.
In a small sauce pan over medium heat, warm 1 tbsp of the Rice Vinegar (2 tablespoon) for a minute. Turn the heat to low and add half of the roasted garlic – miso butter. Stir until warmed through, but not melted. Set aside and keep warm.
In a large skillet over medium-high heat, heat 1 tbsp of olive oil. Add asparagus to the skillet. Cook for 2 minutes, turning occasionally. Add 1 tbsp of the roasted garlic miso butter. cook the asparagus for another minute, until tender.
Add a pinch of salt, give them another toss and then transfer them to a plate lined with paper towels.
While the asparagus is cooking, fill a deep saucepan with water and the remaining tablespoon of rice vinegar. Bring to a boil. Meanwhile, crack the Large Egg (2) into separate bowls.
Once the water is at a boil, turn off the heat. Using a spoon, swirl the water in one direction creating a whirlpool. Holding the bowl of eggs just above the surface of the water, carefully slide one egg into the center of the whirlpool.
Do this again with the second egg. Cover the pan with a lid and let it cook for 3 minutes (no peeking!) Once cooked, use a slotted spoon to gently lift the eggs out of the water. Transfer to a paper towel to drain.
Divide the warmed roasted garlic miso butter between two plates. Add the asparagus and top each with an egg. Season with freshly ground black pepper, and serve immediately.