It seems every chef and many home cooks have strong opinions about the best way to prep mushrooms. They might be scandalized to know that sometimes I rinse my mushrooms instead of cleaning them one by one with a towel. I just don't have the patience. So shoot me! And don't get me started on the idea that mushrooms are best when they're well browned and crispy. I like 'em tender and juicy, as in this recipe. I cook the mushrooms undisturbed in a covered skillet, a technique I learned from my friend Marcia Kiesel, who worked with me for many years at Food & Wine. It sounds bizarre, but it's genius. At first the mushrooms steam in their own liquid; then, as the moisture evaporates, they brown and crisp ever so slightly.
In a large nonstick skillet, melt the butter in the olive oil over medium-high heat. Add the mushrooms and a generous pinch each of salt and pepper. Toss quickly, cover, and cook, undisturbed, until the mushrooms are tender and lightly browned on the bottom, about 8 minutes. Transfer to a medium bowl, stir in the lemon juice, and season with salt and pepper. Cover with plastic wrap to keep warm.
Wipe out the skillet. Fill the skillet with water and heat over medium heat until tiny bubbles appear on the bottom. Stir in the vinegar and a generous pinch of salt. One at a time, crack the eggs into a small bowl and very carefully slide them into the water. Poach over medium heat until the whites are just firm and the yolks are runny, 3 to 5 minutes. Using a slotted spoon, transfer the eggs to a plate.
Meanwhile, toast and butter the bread and then transfer to plates. Stir the basil into the mushrooms and spoon onto the toasts. Carefully slide the poached eggs onto the toasts over the mushrooms. Drizzle with olive oil and sprinkle with salt and pepper. Serve with a knife and fork.
Do It Ahead: The cooked mushrooms can be refrigerated in an airtight container overnight. Reheat them gently and stir in the basil before serving.