To make the filling, in a food processor, blend the shrimp into a smooth paste, then add the carrot, ginger, sesame oil, salt, and white pepper, and mix until fully combined.
To pipe the filling, transfer the shrimp mixture into a pastry bag or a Ziploc bag and snip off the tip to portion evenly.
To assemble the egg rolls, lay the egg roll wrapper flat with a corner pointing toward you, pipe a line of filling near the bottom corner, fold the bottom corner over the filling, fold in both sides, then roll upward and seal the top corner with the beaten egg, keeping the wrap snug but not tight.
To fry, in a pot over medium-high to high heat, heat the cooking oil to 350°F (177°C), then carefully add the egg rolls and fry for about 4 minutes, turning occasionally, until golden and crispy.
To serve, remove the egg rolls and drain on a rack or paper towel, then serve hot while crispy.