In a large mixing bowl, combine the cake mix, eggs, vegetable oil, and water. Use a whisk to blend everything until the mixture is smooth and well combined.
In another bowl, beat together the softened butter, cream cheese, vanilla extract, and heavy cream until creamy. Gradually add in the powdered sugar, adjusting the sweetness to your liking.
Place an egg roll wrapper on a clean surface with one corner pointing towards you. Spoon about 2 tablespoons of the cake batter onto the center of the wrapper, followed by a dollop of the cream cheese filling. Fold the corner closest to you over the filling and roll it up tightly. Brush the edges with the lightly beaten egg to seal. Repeat this process with the remaining wrappers and filling.
In a frying pan or deep fryer, heat the olive oil over medium heat. Once hot, carefully add a few egg rolls at a time, frying them until they are golden brown and crispy, about 2-3 minutes per side. Use a spatula to flip them as needed.
While they are still warm, mix the granulated sugar and cinnamon in a shallow dish. Roll the fried egg rolls in this mixture until they are fully coated.
Place the Birthday Cake Egg Rolls on a cooling rack to let excess oil drip off. Drizzle with chocolate sauce and sprinkle with colorful sprinkles before serving for that festive touch!