In the bowl of a stand mixer fitted with the dough hook, stir together warm milk, yeast, and sugar. Let stand until creamy, about 5 minutes.
Meanwhile, in a medium mixing bowl, whisk together 3 1/2 cups flour, cinnamon, nutmeg, allspice, and salt. Stir in currants. Set aside.
Return to the yeast mixture and add 2 Tbsp softened butter and about half the dry ingredients. Mix on low until the ingredients begin to come together.
Add remaining dry ingredients and mix again on low until flour is fully incorporated.
Increase the speed to medium and knead until the dough is smooth and elastic, about 5 minutes. (The dough will be very sticky and soft at the beginning of the mixing time, but should start to come together. You can add more flour a bit at a time as needed to help it come together, but wait until after it has been mixing for about 4 minutes before adding it. The key to light fluffy rolls is adding as little extra flour as possible.)
Remove the bowl from the mixer and scrape down the sides so that the dough gathers in the bottom. Cover with a dish towel. Let the dough rise in a warm place until doubled in volume, about 20 minutes. (I like to turn on the oven's light and let the dough rise inside the oven. The light produces just enough heat to help the dough rise quickly. In a cool room, it will take about 30 minutes to double.)
Lightly grease a 9×13 baking pan.
Turn the dough out onto a floured surface and form it into an even ball. Dust the top of the dough with some more flour.
Using a serrated knife, cut the dough into quarters. Slice each quarter of dough into 3 even pieces (you should have 12 pieces of dough).
Dust your hands with flour to prevent sticking and roll each piece of dough into a ball, tucking the ends under to form a smooth surface. Place dough balls in even rows in prepared pan.
Heat oven to 400°F / 200°C. Cover the rolls with a dish towel and put them near the warm oven to rise for 20 minutes.
Meanwhile, whisk together 1/4 cup flour and 3 Tbsp of water until it forms a paste similar to thick pancake batter. (Add more water a bit at a time if needed to get the right consistency.) Transfer the flour paste to a small piping bag or sealed plastic bag.
When rolls have finished their second rise, cut a tiny corner off the bag with the flour paste. Pipe a line of flour paste across each row of buns and then pipe a line going the other way to form crosses over the top of each bun. (Note: You likely won’t need all of the flour paste. Discard any that is left.)
Bake until golden brown, 24 to 30 minutes. (Rolls are done when they reach an internal temperature of 190°F / 88°C. If using a metal pan, the rolls should finish near the lower end of the range; if using a ceramic or glass pan, the rolls should finish near the upper end.)
If you want to add the apricot glaze, warm the jam in the microwave for about 15 seconds to soften. Brush warm jam over top of the warm rolls.