To Roast the Asparagus:
Adjust the oven rack to the middle position, and preheat the oven to 450 degrees.
To check the asparagus for tenderness, bite into the end. If it’s stringy and tough, peel the outer layer of the bottom inch or two of each stalk. Place the asparagus in a large bowl, drizzle the olive oil over it, and sprinkle with salt, tossing to coat. Scatter 16 of the thyme sprigs onto a baking sheet and lay the asparagus stalks over them to cover. Roast in the oven for about 10 minutes, until tender to the touch yet firm in the center.
Brush the bread with olive oil and lightly brown each side in a heavy-bottomed pan or, better yet, a well-seasoned cast-iron skillet. Rub one side of each slice of bread with the garlic clove, and place on a baking sheet, garlic side up. Arrange the asparagus over each slice of bread and place on a baking sheet. Adjust the oven rack to the upper position, and preheat the broiler.
To Poach the Eggs:
In a medium saucepan, bring 2 quarts of water to a boil. Turn the heat down to just below a simmer and add the vinegar and salt. Crack an egg into a small bowl to check that the yolk isn’t broken. Slowly stir the water in one direction to create a whirlpool effect. Once the whirlpool has slowed down a little, carefully slide the egg into the water against the side of the pan, following the current of the water as you pour the egg in, so that the white envelops the yolk. Crack another egg into the small bowl and add to the water in the same manner. Poach the eggs for 2 to 2 ½ minutes, until the whites are set and the yolks are runny. Carefully, remove with a slotted spoon to a plate to drain. Cupping the eggs in your hand, tip the plate to pour off the excess water.
Place the eggs centered over the asparagus. Poach the other 2 eggs in the same manner. (If you want to prepare the eggs ahead of time, slip the poached eggs into a bowl of ice water, and just before assembling the sandwiches, return them to the simmering water for 10- 20 seconds to warm.)
Drape a slice of Fontina cheese over each egg, and heat the sandwiches under the broiler for a minute or so, just until the cheese is melted.
Remove the sandwiches from the oven, and rumple a slice of prosciutto over each end of the asparagus, allowing the ends to peek out. Drizzle a little olive oil over, sprinkle with pepper, and place a thyme sprig on each sandwich.