Bring 3 quarts of water to a boil and add 1 T of salt. Set an ice bath nearby. Blanch the asparagus spears in boiling water for 1-1/2 minutes, then remove and immediately refresh in the ice bath. Once cooled, remove from the ice and pat dry.
In a sauté pan, heat the butter over high heat until the foam subsides, add the thyme, and lower the heat to medium, continuing to cook the butter until it browns. Add the asparagus and toss in the butter over high heat for 1 minute, season with salt and freshly ground black pepper.
In a clean sauté pan add the olive oil and heat over medium high heat. Crack 2 of the eggs into the pan, taking care to keep the yolks intact. If you use a mold, remember to oil it first. Cook the eggs sunny side up until the whites are firm but the yolk is still runny. Season each egg with salt and pepper, then carefully slide each egg onto 2 of the asparagus plates. Cook the remaining 2 eggs in the same way.
Divide the brown butter and cooked asparagus evenly among four warmed plates. Spoon remaining browned butter over each of the eggs, grate the cheese over each plate, and serve immediately.