2 sprigs thyme, plus picked thyme leaves for serving
⅓ cup half and half
2 slices of 1 ½-inch thick brioche, white country loaf, or challah bread
Kosher salt and freshly ground black pepper to taste
Heat 2 tablespoons of the butter in a 10-inch skillet over medium heat. Add the garlic and cook, stirring, until softened and fragrant, 1–2 minutes. Add the mushrooms and thyme sprigs, season generously with salt and pepper, and cook, stirring, until the mushrooms soften and release their water, 5–6 minutes. Add the half and half and cook, stirring, until the sauce thickens slightly, 2–3 minutes. Remove the thyme sprigs and season with salt and pepper to taste. Remove from heat and cover to keep warm.
Use a cookie cutter or a glass to cut a 3-inch hole out of the center of each piece of bread. Butter each side of each slice of bread with 1 tablespoon of butter and season with salt. Heat a 10-inch nonstick skillet over medium heat, then add the bread slices and cook until the bottom side is crisped and golden, 3–4 minutes. Flip, then add ½ tablespoon of the remaining butter into the center of each piece of toast. Crack 2 eggs into each piece of bread and cook until the whites are mostly set and the yolks still appear raw, 3–4 minutes. Use a large spatula to carefully but swiftly flip each egg-filled toast and cook an additional 30 seconds.
Divide the mushroom sauce among two plates, then top each with a piece of egg-filled toast. Spoon some of the sauce over the top of each toast and garnish with thyme and cracked black pepper.