Fill a high-sided, 5-quart pot halfway with canola oil. Preheat the oil over medium heat to 350 degrees F
Sprinkle the shrimp on both sides with salt and pepper. Coat a large saute pan with enough oil to coat the bottom and place on medium heat. Add the shrimp and saute, flipping once, until they begin to curl and turn opaque, 1 to 2 minutes. Add the garlic to pan halfway through and cook briefly, making sure not to burn. Deglaze the pan with the white wine and cook off the liquid. Remove from the heat and pour the contents onto a tray to cool.
Bring the pan back to the heat and add a little more oil. Add the Brussels sprouts, cook to sweat and then season with salt and pepper. Add the chiles and cook for another minute. Remove from the heat and set aside to cool. Place the cooked shrimp on the cutting board and run knife through it for a rough chop. Place into a mixing bowl. Mix in the Brussels sprouts, chopped peanuts and Thai basil. Season with salt and pepper.
To build the egg roll, lay the wrapper flat on the work surface like a diamond, with one corner facing you. Brush a little egg wash over the top corner and edges of the wrapper. Lay some of the stuffing in the bottom third of the wrapper in a horizontal log, leaving a 1-inch gap on either side. Fold the bottom corner over the stuffing, then fold the sides inside like an envelope and roll all the way to the top. Place the egg rolls on a sheet tray until ready to fry.
Fry the egg rolls in the oil until golden brown and crispy, 3 to 4 minutes. Remove from the oil carefully and place them on a paper-towel-lined sheet tray. Place on a platter and serve with the Sesame Ginger Dipping Sauce.
In a small bowl whisk together the soy sauce, honey, scallions, vinegar, ginger, oil and chiles. Add salt to taste.