Measure all ingredients. Cut the butter into smaller pieces. Separate the egg yolks from the whites. Grate the yellow part of the lemon zest.
Add 3 cups all-purpose flour and ½ teaspoon salt to a large mixing bowl and stir to combine. Using a spoon, make a shallow well in the center.
Warm ¾ cup milk to 100–110 °F (40 °C). Pour about ½ cup of the warm milk into the well. Add ½ teaspoon of the sugar and 1 ½ teaspoons active dry yeast. Using a fork, lightly mix the liquid with a little flour to form a semi-liquid puddle.
Let sit in a warm place for about 15 minutes, until the yeast becomes foamy and bubbles appear on the surface. If no bubbles form, the yeast is not active. Start again with fresh yeast.
Meanwhile, gently rewarm the remaining milk. Add ½ stick unsalted butter and let it melt slowly. Keep the heat low to medium so the mixture does not become hot. Remove from the heat and whisk in 2 Tablespoons spiced rum, 2 teaspoons lemon zest, and the rest of 3 Tablespoons granulated sugar, then whisk in 3 egg yolks.
Pour the warm, but not hot, mixture into the bowl with the flour and activated yeast. Mix with a fork or wooden spoon just until a rough dough forms.
Knead the dough in a stand mixer fitted with a dough hook on medium speed for 8 minutes, until smooth, elastic, and barely sticking.
Transfer the dough to a large clean bowl. Cover with plastic wrap and let rise in a warm, draft-free place for about 40 minutes.
Gently lift the dough a few times with your fingers to release excess air and flatten it slightly. Do not punch it down. Cover again and let rise for another 20 minutes.
Turn the dough out onto a lightly floured work surface. Roll to about ½ inch (1.5 cm) thickness. Cut out circles about 3 inches (7.5 cm) in diameter. Cover with a clean kitchen towel and let rest for 20 minutes.
Gather the scraps, gently press together, shape into a ball, lightly spritz with water, cover, and let rest.
Heat 4 cups canola to 350 °F (180 °C). Use a thermometer and keep the temperature steady.
Fry the rested dough circles for about 1½ minutes on one side, flip, and fry for another 1½ minutes on the other side, so they are golden on the surface.
Remove with a metal slotted spoon and transfer to a cooling rack lined with paper towels.
Re-roll the remaining rested dough, cut out the circles, and fry them the same way.
Once slightly cooled, fill the koblihy with jam from the side using a piping bag or pastry injector.
Dust generously with powdered sugar before serving.