Bring a medium saucepan of water to a boil; decrease heat so the water simmers gently.

Using a slotted spoon, gently lower the eggs into the water; cook for 11 minutes and then let cool.

While the eggs are cooking, grill the asparagus on a preheated panini grill; grill them for 1 minute until they are warm but still have a degree of crispness to them.

Transfer to a cutting board and slice on the bias, about ¼ inch thick.

Peel the eggs from their shells; use your fingers to break the eggs in half, transferring the yolks to a mixing bowl and the whites to a cutting board.

Break the yolks with a fork, mashing until they resemble cornmeal; add in the sliced asparagus.

Roughly chop the whites into small but uneven pieces; transfer to mixing bowl; add in the mayonnaise.

Gently fold the ingredients together; the egg salad should have a creamy consistency, so add a tablespoon or two more mayonnaise, if necessary; season to taste with salt and pepper.

Place half of the white bread slices on a clean work surface; spread about ¼ cup egg salad over each and top with remaining bread slices.

Before serving, use a serrated knife to remove the crusts from the sandwiches and cut into triangles.