Place eggs into a large pot and cover with water. Bring to a rolling boil, then turn off heat and cover with a lid. Let eggs set for 15 minutes until cool to the touch then peel.
Thinly slice eggs. Using a small, round food cutter, cut bread into smaller pieces. (Aim for 2 shapes cut out of each slice of bread.) Mash avocado and mix with lime juice.
Assemble little sandwiches: spread mashed avocado onto bread, then add spinach leaves and egg slices. Serve with grapes on the side.