Warm the Whole Milk gently (about 38–43°C / 100–110°F) and stir in 2 tablespoons of Sugar until dissolved; sprinkle over the Instant Dry Yeast with a tablespoon of Water and let sit 5–10 minutes until foamy.
In a large bowl whisk together the All-Purpose Flour, a pinch of Salt and the grated Lemon Zest.
In a separate bowl beat the Eggs with the Egg Yolk, then stir in the Vanilla Extract and Dark Rum.
Combine the yeast-milk mixture and the egg mixture with the flour, then add softened Unsalted Butter and knead (by hand or in a stand mixer) until the dough is smooth, elastic and slightly tacky; add a little more Water if the dough feels too stiff.
Place the dough in a lightly oiled bowl, cover, and let rise in a warm place until doubled in size, about 1–1½ hours.
Turn the risen dough onto a lightly floured surface, roll to about 1.25 cm (½ inch) thickness and cut into rounds (approximately 7–8 cm / 3 inches). Place the rounds on a tray, cover lightly and let them proof until puffy, about 30–45 minutes.
Heat enough Neutral Frying Oil in a heavy pot to reach about 170°C (340°F). Fry the proofed rounds a few at a time, turning once, until deep golden and cooked through — roughly 2–3 minutes per side (adjust time slightly for size and exact temperature).
Drain the fried Berliners on paper towels and let them cool just enough to handle but still warm.
Fill each bun with Jam using a piping bag fitted with a long tip (or spoon the jam into a small hole you make with a skewer), injecting about a tablespoon or to taste.
While still slightly warm, toss or dust the filled Berliners in remaining Sugar so the exterior is evenly coated. Serve warm for best texture and flavor.
