Combine golden raisins and rum in a small bowl and let soak.

Dissolve yeast and 1/2 teaspoon sugar in 1/2 cup milk in a bowl. Let stand until yeast softens and begins to form a creamy foam, about 10 minutes.

Mix remaining 1 cup milk and 1 cup flour together in a bowl. Add yeast mixture and stir well. Cover and let stand in a warm place for 30 minutes.

Beat 5 egg whites and salt in a glass, metal, or ceramic bowl until stiff peaks form.

Combine remaining sugar and egg yolks in a large bowl; beat with an electric mixer until fluffy and pale, about 5 minutes. Beat in vanilla extract. Drain raisins and add rum to egg mixture. Add to yeast mixture and mix well. Fold in egg whites gently. Add remaining flour in small batches, mixing well after each addition, until dough starts to pull away from the sides of the bowl.

Turn out dough onto a floured work surface and knead for about 10 minutes, adding more flour as needed. Start rubbing in 1 cup plus 2 tablespoons butter, 1/4 cup at a time, once dough seems pliable and soft. Knead for 2 more minutes and form into a ball.

Transfer to a clean, lightly oiled bowl. Cover with plastic wrap and a clean dish towel. Allow to rise in a warm place until doubled in volume, about 90 minutes.

Turn dough out onto a floured work surface and knead for 2 minutes. Sprinkle flour over raisins. Scatter raisins, almonds, and mixed peel in a circle around the dough. Keep kneading dough to incorporate fruits and nuts evenly.

Line 2 large metal coffee tins with parchment paper. Divide dough in half and transfer to the tins. Cover with a clean dish towel. Allow to rise in a warm place until dough has risen to the top, 30 minutes to 1 hour.

Preheat oven to 350 degrees F (175 degrees C).

Bake kulich in the preheated oven until top is golden brown and a toothpick inserted in the middle comes out clean, about 1 hour. Carefully tip loaves out of the tins onto a work surface and let cool, about 2 hours.

Combine confectioners' sugar, 1 egg white, and lemon juice in a small bowl. Brush over the the tops of the kulich loaves, allowing icing to run down the sides.