Preheat the oven to 350 degrees F. Line a baking sheet with parchment paper. Place 4 to 5 asparagus spears in a row on one corner of the baking sheet; season with salt and pepper. Top asparagus with 2 slices of cheese and 1/4 cup crabmeat. Repeat with remaining asparagus, cheese and crab. Bake 6 to 8 minutes, or just until the cheese begins to melt. Remove from the oven.
Spread croissants with mayonnaise. Top bottom halves with salad greens. Set 4 to 5, cheese- and crab-topped asparagus spears on each croissant. Set on top half of each croissant and serve.
Note: To blanch asparagus, plunge into boiling water for 1 to 2 minutes. Cool in ice cold water, and then drain well.