1. In a 12-inch skillet, heat 6 teaspoons oil over medium-high heat. Add mushrooms and onion and cook, stirring occasionally, until liquid has evaporated, about 10 minutes. Add garlic and cook, stirring, until fragrant, about 2 minutes. Add tomato paste, thyme, salt, pepper and wine and cook until liquid is absorbed, about 3 minutes. Add kale and cook, tossing to combine, until wilted, about 2 minutes.
2. Using a spoon, create four large indentations in the mushroom-and-kale mixture and drop an egg into each. Cover skillet with aluminum foil and cook until eggs have just set, about 3 minutes.
3. Meanwhile, turn on the broiler. Arrange bread on a baking sheet and drizzle each slice with teaspoon oil. Broil 2 minutes, or until just browned; rub with garlic and serve with ragout.