The day before baking, mix flour, cold water, and sourdough starter together in a bowl. Cover and leave out at room temperature, 8 hours to overnight.
At the same time, combine raisins, cherries, and pineapple in a separate bowl. Stir in rum. Let fruit soak, tossing occasionally, 8 hours to overnight.
Pour warm water into large bowl of a stand mixer fitted with a dough hook attachment. Add yeast and let dissolve for 10 minutes. Add eggs, 1/3 cup plus 2 tablespoons sugar, vanilla extract, orange zest, and lemon zest. Whisk together. Scoop in the panettone starter. Add flour and salt.
Knead until dough is very smooth and elastic, about 10 minutes. Stop and scrape down the sides if needed. Add butter; knead until butter is completely mixed in and dough is soft and somewhat sticky, about 5 minutes. Transfer to your work surface.
Toss and fold your dough into a rough ball shape using wet fingers and a bench scraper. Transfer dough back to the bowl. Cover and let rise until doubled, about 3 hours.
Transfer dough back to your work surface and shape into a rough ball. Place into a resealable plastic bag. Refrigerate, 8 hours to overnight.
Remove dough from the bag and press out into a flat rectangle. Sprinkle with flour and roll out to at least 1/2-inch thickness. Spread the dried fruit all over the surface. Roll dough up into a log, sealing in the fruit. Roll both ends of the log toward the middle and smooth it over into a ball.
Place dough ball into a short, wide paper panettone mold. Cover with plastic wrap and let rise until dough is at least 2/3 of the way up the sides, 3 to 4 hours.
Preheat the oven to 350 degrees F (175 degrees C).
Mix egg and water together and brush mixture on top of the panettone. Cut 2 slits crosswise on top.
Bake in the preheated oven until beautifully browned, 40 to 45 minutes. Poke 2 skewers through either side of the panettone and flip it over into a panettone hole or a Dutch oven. Let cool upside-down for 2 hours.