Cook the bacon (try one of my easy methods listed below). Set aside.
Next, hard boil the eggs. In a small pot, bring just enough water to cover the eggs to a gentle boil. Reduce the heat to a simmer and gently lower the eggs in with a slotted spoon. Bring water to a low, rumbling boil, then let cook for 9 minutes. A few minutes before the eggs are finished cooking, prepare an ice bath. Remove the eggs from the boiling water and plunge into the cold water.
Let rest a few minutes, then tap the eggs with the back of a spoon to crack them (this will make peeling easier). Return to the ice water to cool completely. Peel the eggs, then cut in half and discard the yolks from three whole eggs (six halves), so that you have two whole eggs and three with egg whites only. Roughly chop and place into a large mixing bowl.
Halve and pit the avocados, then with your knife, gently make a criss-cross pattern to divide the flesh into squares.
Scoop the flesh into the mixing bowl, leaving the avocado in chunks. Add the chopped green onions, Greek yogurt, lemon juice, mustard, salt, pepper, and cayenne to the bowl.
Fold together, leaving the eggs and avocado chunky.
Toast the bread, then assemble the sandwiches with bacon, avocado egg salad, sliced tomatoes, and spinach. Enjoy immediately.