Stir together the yeast, 1 tsp. of the sugar, and the warm water in a large bowl, and let sit until bubbling and frothy, 5 to 10 minutes. Meanwhile, whisk together 3 cups of the flour, the remaining 1/4 cup sugar, the cardamom, and salt in a separate bowl. Stir the egg yolks, milk and vanilla into the yeast mixture. Add the flour mixture in two additions, stirring until the dough begins to come together.
Turn the dough out onto a lightly floured surface and knead, adding up to 1 cup of flour, a little at a time, until the dough is smooth and supple, 5 to 10 minutes. (You may not need all of the flour.) The kneading can also be done in a stand mixer fitted with a dough hook for 5 to 7 minutes.
Rub about 1 tsp. of vegetable oil around the large bowl; place the dough in the bowl and turn to coat. Cover the bowl with a kitchen towel and let the dough sit in a warm place until doubled in size, about 1 1/2 hours.
Line a large plate with paper towels. Add oil to a medium saucepan until it's about 1 1/2 inches deep and set the pan over the medium heat until the temperature reaches 365 on a candy or deep-frying thermometer. Working in batches of five or six, pinch off walnut-size pieces of dough, roll each into a ball shape, and drop into the hot oil. Fry, flipping once, until puffed, golden and cooked through, 2 to 4 minutes total. Transfer with a slotted spoon to the prepared plate to drain.