1. Put eggs in a medium pot and cover with cold water. Bring to a boil, then cover, remove pot from heat, and let sit 14 minutes. With a slotted spoon, transfer eggs to a bowl of ice and cold water (reserve hot water in pot). Cool eggs at least 10 minutes, then use a slotted spoon to transfer eggs back to hot water (to loosen shells) for 10 to 20 seconds. Pat dry and peel.
2. Meanwhile, cut cucumbers into 24 rounds (3/4 in. thick each; you may have some cucumber left over). Using the size of biscuit cutter that fits best (see Notes), cut out fluted rounds. With a melon baller or small spoon, scoop seeds from centers of cucumber rounds, leaving at least a 1/4-in layer of seeds and/or flesh. Set on a baking sheet.
3. Shred eggs with a razor-sharp grater (such as a Microplane) or against the finest slatted (not punched-out) holes of a box grater and put in a medium bowl. Add mayonnaise, red onion, 1 tsp. lemon juice, and 1/2 tsp. salt. Mash mixture together with a fork.
4. Fill cups with egg mixture, mounding slightly. Sprinkle each with salt and a few drops of lemon juice to taste and garnish with a sliver of serrano.
Note: Nutritional analysis is per cucumber cup.