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1
Preheat oven to 450.
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2
Scatter thyme sprigs on a baking sheet.
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3
Lay the asparagus over the thyme sprigs, drizzle with 1 tablespoon of the olive oil and sprinkle with salt.
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4
Roast in the oven for about 15 minutes.
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5
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.
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6
In a small saucepan, cook the butter over medium-high heat until the bubbles subside and the butter is browned with a nutty aroma.
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7
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.
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8
Rub one side of the toast with the garlic.
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9
Lay the toasts on plates and top with the roasted asparagus.
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10
In a 4-inch cast-iron or heavy-duty skillet, heat 1/2 cup of the olive oil until very hot but not smoking.
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11
Crack an egg (or 2 eggs) and add to the hot oil.
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12
Sprinkle with fleur de sel and cook for about 1 minute, until the whites are bubbly and the outer edges become brown and crisp.
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13
Just before the eggs are done, spoon a little of the oil onto the yolks to create an opaque film over them.
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14
Remove the eggs and place over the asparagus.
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15
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.
