1. Slice the garlic in half and rub the inside of a medium bowl with each half. Then grate the garlic clove into the bottom of the bowl.
2. To coddle an egg for the dressing, bring a medium saucepan of water to a gentle boil over medium heat. Drop in 1 egg and set a timer for 1 minute. Remove the egg with a large slotted spoon, and then crack the egg to separate the yolk from the white, discarding the white and the shell. Keep the pot with the water for later use.
3. Add the yolk to the bowl with the garlic, and then add the Dijon and anchovies. Slowly stream in the olive oil and whisk until the dressing thickens. (It will be very thick at this point, but then will thin out as you add the lemon juice.) Add the lemon juice and season with salt and pepper. Set aside until ready to assemble the sandwiches.
4. Fill a large bowl with lots of ice and some water.
5. Make sure the saucepan you used to boil the first egg has enough water in it to cover 4 more eggs. Add the white vinegar to the pot and bring to a boil. Using a large slotted spoon, gently lower the eggs into the water, and then reduce the heat to medium-low. Set a timer for 7 minutes. Once the timer goes off, use the slotted spoon to remove the eggs and place into the ice bath to stop the cooking, at least 2 minutes. Then gently peel and set aside.
6. Lightly toast the bread slices and spread with the softened butter.
7. To build a sandwich, spoon a generous amount of the anchovy aioli (about 2 tablespoons) over a toasted bread slice, and then layer on a few tomato slices. Top with a few avocado slices, squeeze a bit of lemon juice on the avocado and season with salt and pepper. Take a paring knife and pierce an egg in the center. Then hold the egg over the sandwich, gently tear it in half and place on top of the avocado. Season with salt and pepper. Repeat with the remaining ingredients. Serve any leftover aioli in a bowl for dipping.