Prepare Dough: In small bowl or 1-cup measuring cup, combine water, yeast, and 1/2 tablespoon sugar; stir to dissolve yeast. Let stand until foamy -- about 5 minutes.
In medium-size bowl, combine 1 1/2 cups flour, the remaining 3 tablespoons sugar, and the salt; stir in yeast mixture, eggs, and oil, mixing until combined.
Turn dough out onto floured surface. Knead dough, adding as much of remaining bread flour as necessary to prevent stickiness, until smooth and elastic -- about 8 minutes.
Wash, dry, and oil mixing bowl; place dough in bowl and turn to bring oiled side up. Cover with clean cloth and let dough rise in warm place, away from drafts, until double in size -- about 1 1/2 hours.
Meanwhile, prepare Filling: In large skillet, heat oil over medium heat; add gingerroot and sauté 1 minute. Add garlic and cook 30 seconds. Stir in beef, hoisin sauce, soy sauce, paprika, and ground red pepper. Cook until beef browns. Remove from heat and stir in green onions and sweet red pepper. Set aside.
Grease 2 baking sheets or line with parchment paper. When dough has doubled, punch down and shape into a 10-inch-long log. With serrated knife, cut log crosswise into ten 1-inch-thick slices. On lightly floured surface, roll out one slice to a 4-inch round. Roll out or press just around the edge of round so that the middle is slightly thicker and the round is 5 inches wide. Fill center of round with 1 heaping tablespoon beef filling; gather up edge to form a pouch and pinch together tightly to seal bun. Turn bun over and gently shape into a ball; place seam side down on greased sheet. Repeat with remaining slices and filling. Cover beef buns with clean cloths or plastic wrap and let rise 30 minutes. Meanwhile, prepare Egg Glaze.
Heat oven to 350 degrees F. Brush buns with Egg Glaze and sprinkle, if desired, with sesame seeds. Bake 20 to 25 minutes or until well browned. Cool buns on wire rack; serve warm.
Egg Glaze: In small bowl, beat together 1 large egg and 2 tablespoons milk until blended; refrigerate when not using.