Place a small amount of water into a saucepan and heat over medium; meanwhile, place a medium-sized glass full of ice water beside the stove. Gently lower the eggs into the boiling water with a spoon.
Reduce heat as needed to maintain a constant temperature. Simmer for 7 minutes, stirring gently occasionally.
Place the avocado halves, sliced down, on a cutting board.Next, cut each half into strips, about 1/4-inch thick. Fan flatten the slices with your fingers and gently press each half down and to the sides.
Carefully place the shingled avocado on the toast, placing 1 slice of avocado on each slice of toast. Sprinkle each with salt and 1/8 teaspoon pepper.