Add 1 tablespoon butter to a medium pot over medium-high heat. Once the butter is melted and bubbling, add the barley and sauté until the grains are fragrant about 2 minutes. Add the water and increase the heat to high until the water comes to a boil. Stir the barley once, reduce the heat to low, cover, and cook until tender, about 25 minutes. Drain any excess water, return to the pot, and stir in the cheese and black pepper. Cover until ready to serve.
Preheat the oven to 425°F. Break up the broccoli and cauliflower into a single layer on a large baking sheet. Bake for 10 minutes, remove and drizzle with the olive oil, and stir to combine. Return to the oven and roast until the vegetables get brown and crispy at the edges, another 30 minutes, stirring halfway through the cook time.
Meanwhile, melt the remaining tablespoon butter in a large skillet over medium-high heat. Add the mushrooms and cook, stirring only 2 or 3 times, until the mushrooms are deeply browned, about 10 minutes
Also meanwhile, boil and peel eggs. A runny egg adds extra sauce to this bowl, so 7 to 9 minutes is the sweet spot.
Divide the barley among four bowls. Top with an equal amount of the broccoli, cauliflower, and mushrooms. Place an egg on each serving. Serve immediately with hot sauce on the side.