How to Make It
Preheat oven to 450°. Scatter thyme sprigs on a baking sheet. Lay the asparagus over the thyme sprigs, drizzle with 1 tablespoon of the olive oil and sprinkle with salt. Roast in the oven for about 15 minutes.
Turn the oven down to 400° F. Spread the guanciale in a single layer in a sheet pan and cook until lightly crisp, 8 to 10 minutes for guanciale or 12 to 15 minutes for pancetta.
In a small saucepan, cook the butter over medium-high heat until the bubbles subside and the butter is browned with a nutty aroma.
Toast the bread in a toaster, or grill it in a sandwich press, making sure that both sides are brushed with extra-virgin olive oil or softened unsalted butter first. Rub one side of the toast with the garlic. Lay the toasts on plates and top with the roasted asparagus.
In a 4-inch cast-iron or heavy-duty skillet, heat 1/2 cup of the olive oil until very hot but not smoking. Crack an egg (or 2 eggs) and add to the hot oil. Sprinkle with fleur de sel and cook for about 1 minute, until the whites are bubbly and the outer edges become brown and crisp. Just before the eggs are done, spoon a little of the oil onto the yolks to create an opaque film over them. Remove the eggs and place over the asparagus. Spoon about a tablespoon of brown butter over the asparagus and egg, arrange the guanciale or pancetta on either side and sprinkle with grated Parmesan cheese.