Arrange the eggs in a saucepan and add enough cool water to cover them completely. Bring the water to a boil, then cook the eggs for 20 minutes.
Transfer the cooked eggs to a bowl of cold water and allow them to rest for 2–5 minutes to halt the cooking process. Once sufficiently cooled, carefully remove the shells.
Cut each peeled egg lengthwise into two equal halves. Gently lift out the yolks, keeping the egg whites intact.
Press the yolks through a fine-mesh strainer into a mixing bowl to create a smooth, finely textured base.
Add the mayonnaise, spicy brown mustard, salt, and pepper. Blend thoroughly until the mixture reaches a dense, hummus-like consistency.
Stir in the red food coloring or beet juice until the filling is evenly tinted. Cover the mixture or place it in a pastry bag so it does not dry out while the remaining components are prepared.
When using sandwich bread rather than rice crackers, preheat the oven to 350°F (175°C).
Cut 12 small rounds from the three bread slices with a round cookie cutter. Flatten each circle into a thin wafer, arrange them on a baking sheet, and toast for 5–7 minutes, or until crisp.
Omit this step when using rice crackers.
Using the end of a pastry tip, cut a small opening in the rounded top of every egg-white half. Remove each cut-out portion and divide it into two pieces.
Return one half of each trimmed piece to its opening, forming a small support for the olive.
Cut the cocktail olives crosswise into slices approximately ½ inch thick. Carefully press one olive slice into the top of each egg-white half, positioning it over the inserted piece.
Turn each olive-topped egg-white half upside down. Pipe or carefully spoon the prepared yolk mixture into the cavity, avoiding excessive filling that could spill from the sides.
Place one toasted bread round or rice cracker directly over the filling to form a base. Gently turn the assembled egg upright so the olive resembles the pupil of an eye.
Continue the same process until all 12 egg halves have been completed.