Make the dough: Add butter into a medium saucepan and heat until melted. Add the cold milk, vanilla extract, and 1 tablespoon sugar, stir until combined, then take the pot off the heat. Pour the mixture into a medium bowl, make sure the temperature of the liquid is between 100-110°F (38-43°C) (pleasantly warm to your finger) then stir in the yeast (adding yeast into a warm liquid will make them rise faster). After about 1-2 minutes you should see the yeast is starting to 'bloom' (will be frothy/foamy).
Add the egg yolks and the remaining sugar into a large bowl. Whisk for 1-2 minutes until fluffy. Add brandy/rum, salt, lemon zest, and flour. Add the yeast mixture.
Stir with a spoon until roughly combined then start kneading by hand or with a stand mixer using a dough hook attachment. Knead the dough until soft, elastic, and no more sticky. It will take 7 minutes on speed 2 in Kitchen Aid or about 15 minutes by hand. If kneading the dough by hand you can find the mixture too sticky, try not to add additional flour.
After 7 minutes of kneading with a mixer, you will see that the dough starts to clean off the sides of the bowl, it will be soft and a little bit sticky. Scrape the dough from the sides of the bowl with a spatula and knead it by hand for a minute. After this time the dough should be smooth, soft and should no longer stick to your hands. If you have used a kitchen scale to measure out the flour you can trust the recipe and use the exact amount written in the ingredient list but if you’re using measuring cups you have to assess the texture of the dough and add more flour/liquid if necessary.
First rise: Form a smooth dough ball, put it back in the bowl, cover the bowl tightly with plastic foil, and place in a warm place for about 1 - 1 1/2 hours or until doubled in volume (I'm putting it in a turned off oven with a light on).
Transfer the dough to a lightly-floured rolling mat/counter and roll it out to a thickness of 2/3 - 3/4 of an inch (1.7-1.9cm). Cut out rounds using a 2 3/4 to 3-inch biscuit cutter (I used a 2 3/4 which is 7cm). You can knead the leftover dough into a ball, roll it out, and again cut out rounds. You can also simply divide the dough into 70-80g pieces, and roll each piece of dough into a ball.
Second rise: Place pieces of dough on a lightly floured piece of parchment paper or a wooden board, about 2-inches (5cm) apart. Cover loosely with a clean kitchen towel. Let rise for 20-30 minutes.
Fry Krapfen: Heat the oil in a large pot (preferably shallow and wide) until the temperature reaches 350°F (175°C). It's best to use a candy thermometer which can be attached to a pot or a digital grill thermometer to check the temperature for the whole frying process (it should be 340-350°F which is 170-175°C). The amount of oil will depend on the size of the pot (I used 6 cups), you should have at least 1 1/2 inches (4cm) oil in your pot.
Fry Krapfen in batches (3-5 Krapfen, depending on the size of the pot). Gently lower the Krapfen into the oil, I like to pick them up and slide them off of a lightly floured bench scraper.
It’s important to control the temperature of the oil – if it’s too hot, Krapfen will brown too quickly and will be raw in the middle, if it’s too low, they will soak up fat and be dense and greasy. Fry Krapfen for about 2-3 minutes per side or until golden.
Transfer to a plate lined with paper towels. Let them cool down.
When Krapfen are cold, fill them with apricot jam using a piping bag and a long piping tip.
Dust them generously with powdered sugar.