Make the creamy egg filling (whipped-scramble method — fast & reliably creamy)
In a medium bowl, crack 6 eggs. Add 4 tbsp mayonnaise, 1–2 tsp milk or cream (if using), 1/4 tsp salt, 1/4 tsp pepper, and 1/4 tsp sugar (optional). Whisk vigorously until uniform and slightly frothy.
Heat an 8–10" nonstick skillet over low heat. Add a small knob of butter or a little neutral oil and swirl to coat. Turn heat to low — key is gentle, low heat.
Pour the egg mixture into the pan. Stir constantly with a silicone spatula in slow, continuous motions, scraping the cooked edges into the center. Cook low and slow until eggs are just set but still glossy and custardy (about 4–6 minutes). Remove from heat slightly before fully firm to avoid overcooking.
Transfer the eggs to a bowl. Fold gently to break large curds into a smooth, spreadable texture. Taste and adjust salt/pepper. If you want extra creaminess, fold in 1 tbsp more mayonnaise. Let cool slightly (or chill briefly for firmer filling).
Assemble the sandwich (classic tamago sando style)
Remove crusts from bread slices (if desired). Lightly spread a thin layer of softened butter on the inside of each slice (optional — helps prevent sogginess).
Place 4 slices face-up. Spoon an even layer of egg filling onto each, aiming for about 1/2–3/4 inch thickness. Smooth gently with the back of a spoon or an offset spatula. Top with remaining bread slices. Press gently.
For clean edges and even shape, wrap each sandwich tightly in plastic wrap and allow to rest 10–15 minutes (this compresses filling and makes cutting tidy).
Unwrap and slice sandwiches in half (or into rectangles), cutting with a sharp serrated knife using a gentle sawing motion. Serve immediately, or keep chilled until serving.
Quick alternative method (tamagoyaki / rolled omelet method — firmer, layered filling)
Whisk eggs with 1 tbsp water, 1 tbsp sugar (optional), a dash of soy sauce (optional), and 1/2 tsp salt. In a tamagoyaki pan or small nonstick skillet, cook thin layers, rolling each layer to form a log. Press in plastic wrap to shape, slice, then roughly chop and fold with mayo to create filling. Assemble as above.