Using a rolling pin, gently press each slice of trimmed bread until evenly flattened to create a pliable surface for rolling.
Position one sausage link along the short edge of each bread slice, then roll the bread tightly around the sausage. Place seam-side down to hold its shape.
In a shallow mixing bowl, thoroughly combine the eggs, milk, cinnamon, and sugar, whisking until smooth and fully blended.
Briefly dip each assembled roll into the egg mixture, ensuring all sides are evenly coated. Avoid soaking them too long to maintain structure.
Heat a nonstick skillet over medium heat and melt a small amount of butter or oil. Place the rollups seam-side down in the pan and cook, turning every couple of minutes, until all sides are golden brown and crisp, approximately 6 to 8 minutes total.
Serve immediately while warm. Optionally, dust with powdered sugar or extra cinnamon-sugar and offer maple syrup or fruit on the side.