Preheat oven to 425°.
Cover a large, heavy baking sheet with parchment paper; set aside.
Lightly spoon flour into a dry measuring cup; level with a knife. Set aside.
Combine 1 cup water, butter, and salt in a large, heavy saucepan over medium-high heat; bring to a boil, stirring occasionally with a wooden spoon. Add flour, stirring well until mixture is smooth and pulls away from sides of pan. Cook 30 seconds, stirring constantly. Remove from heat. Place dough in bowl of a stand mixer. Add egg whites and eggs, 1 at a time, beating at medium speed with a paddle attachment until well combined. Beat 2 minutes at medium speed.
Scoop the dough by 1/4 cupfuls into 8 mounds 2 inches apart onto prepared pan. Bake at 425° for 20 minutes. Reduce oven temperature to 350° (do not remove pan from oven); bake an additional 30 minutes. Turn oven off. Pierce top of each cream puff with a knife; return pan to oven. Cool cream puffs in closed oven for 20 minutes. Remove from oven; cool completely on a wire rack.
Split cream puffs in half horizontally. Fill each puff with 1/4 cup ice cream and 1 tablespoon syrup.