Place a skillet over medium-high heat. Once hot add oil. Add mushrooms, carrot, and garlic and sauté for 2 to 3 minutes. Remove from heat and cool slightly.
Fill a tall pot with oil and preheat to 350˚F. Place all filling ingredients (including sautéed mushrooms, carrots, and garlic mixture) into a large mixing bowl.
Mix together until very well combined. Refrigerate for at least 30 minutes and up to 1 day.
Place a single egg roll wrapper onto a clean surface (so it looks like a diamond)and brush the surface with a light layer of egg wash.
Take the bottom corner of the wrapper and fold it about ½ way up to the opposite corner and press down to smooth and seal.
Place about 2 tablespoons of filling into a narrow log onto the overlapped portion of the wrapper. Take both sides of the wrapper and fold them in, towards the center of the log of filling.
Brush the ends of the wrapper with more egg wash (as needed) and tightly roll each egg roll until completely closed and sealed.
Place onto a parchment lined baking sheet, seam side-down. Repeat until all the filling and wrappers have been used.
Carefully drop 4 to 5 egg rolls into the hot oil at a time and fry for 5 to 6 minutes or until golden brown and filling has just cooked through. Drain onto paper towels for 1 minute before transferring onto a baking sheet, lined with a cooling rack. Repeat until all egg rolls have been fried.
Transfer to a platter and serve with sweet-chile dipping sauce.