Melt the butter in a skillet over medium heat. Add the mushrooms and sauté until brown, 5 to 7 minutes. Season to taste with salt and pepper to taste.
Add shallots and continue cooking until soft and fragrant, 2 to 3 minutes. Add the white wine and cook, scraping any browned bits from the bottom of the pan, until evaporated. Remove from the heat and cool the mixture for at least 15 minutes.
Combine the goat cheese, chives and cooled mushroom mixture in a bowl until well blended.
Place a single egg roll wrapper with one corner facing you (like a diamond shape) and spoon about 1 tablespoon of the mushroom mixture into the center.
Fold the corner closest to you over the filling, then fold the side corners toward the center. Moisten the top point of the wrapper with a bit of water and roll tightly in a jelly-roll fashion. Repeat the process with the remaining wrappers and filling.
Place each finished egg roll, seam side down on a plate and cover with a damp paper towel until you're ready to fry them.
Heat about 1/4-inch of vegetable oil in a large, heavy skillet over medium-high heat. Working in batches of 3 or 4 egg rolls at a time, add them to the pan, seam side down and cook, turning frequently to achieve an even, crispy brown finish, 5 to 6 minutes total.
To serve, cut each egg roll in half on the bias and serve immediately.