Make the syrup. Heat the syrup sugar and water in a saucepan, stirring only until the sugar dissolves. Bring to a boil, then simmer gently, uncovered, for 10 minutes. Stir in the lemon juice and simmer 1 minute more.
Cool the syrup completely. Remove from the heat and let the syrup reach room temperature with no residual warmth. Transfer to a wide dish on a stable surface well away from the frying pot. Make the dough while it cools; do not start frying until the syrup is fully cool.
Cook the dough. Bring the dough water, dough oil, and salt to a boil in another saucepan. Reduce the heat and add all the flour at once. Stir vigorously over low to medium heat for about 3 minutes, until the dough forms one mass, pulls away from the pan, and leaves a thin pale film on the bottom.
Cool and mix in the eggs. Transfer to the mixing bowl and cool for about 15 minutes, until only lukewarm. Work in the eggs one at a time, completely incorporating each before adding the next. The mixture may separate briefly before coming together. Sift in the baking powder and mix for about 30 seconds. The smooth, glossy dough should hold distinct piped ridges; do not taste raw dough.
Set up for frying. Fit a sturdy piping bag with the open star tip and fill with dough. Put the frying oil into a dry, heavy, high-sided pot, never filling it more than one-third full. Heat to about 329°F (165°C). Monitor with a frying thermometer and maintain 320–338°F (160–170°C). Keep water, wet utensils, children, and the syrup dish away from the hot oil.
Pipe small batches. Lightly oil the scissors with a little cold oil reserved from the frying supply. Hold the tip close to the oil without touching it. Pipe lengths about 2⅜ inches (6 cm) long, snipping each close to the surface to limit splashing. Keep hands back. Fry only a small batch with room to expand and move freely; about six to eight pieces fit the source-size pot, but use fewer if crowded.
Fry until cooked through. Fry for about 6–8 minutes per batch. Wait until the surface sets before turning, then turn carefully several times for an even dark-golden color. Keep the oil within the stated range. Lift a sample onto a clean plate to check the first batch: the center should be fully set, with no wet, gummy or raw dough, and reach at least 160°F (71°C) on a clean fine-tip thermometer. Color and the clock alone do not prove doneness; fry longer as needed at the controlled temperature.
Soak while hot. Lift the cooked pastries out with a slotted spoon and drain on paper towels for only 20–30 seconds. Immediately put them in the completely cooled syrup. Soak for 5–7 minutes, turning once or twice. Use a separate clean slotted spoon to lift them into a shallow dish in a single layer.
Repeat and cool. Bring the oil back to about 329°F (165°C) before each batch. Repeat with the remaining dough, coordinating the brief soaking with the next batch without leaving hot oil unattended. Turn off the heat when finished. Rest the soaked pastries for at least 30 minutes, until fully cool; serve at room temperature or chilled. Refrigerate within the time limits in Notes.