Warm the milk until lukewarm, then crumble in the yeast. Add a teaspoon of sugar and let it proof for about 10 minutes, until foamy.
In a large bowl, whisk together the flour, remaining sugar, and salt. Add the proofed yeast, melted butter, and eggs. Start kneading the dough.
Knead the dough for at least 10 minutes, until smooth and elastic. If it's too sticky, add a little flour; if it's too dry, add a little milk.
Cover with a clean kitchen towel and let rise in a warm place for about an hour, or until doubled in size.
On a lightly floured surface, roll out the dough to about 1 cm (½ inch) thickness. Use a glass or doughnut cutter to cut out circles.
Let the cut-out doughnuts proof for another 30 minutes to make them even lighter during frying.
In a deep skillet or pot, heat the oil to around 340-355°F (170-180°C). Fry only a few doughnuts at a time to avoid cooling the oil down too much.
Fry the doughnuts for 2-3 minutes per side, until golden brown. Remove them to a paper towel-lined plate to drain excess oil.
To make the caramel, melt the sugar in a saucepan over medium heat until it turns amber in color. Carefully pour in the warm heavy cream, stirring constantly.
Add the vanilla extract and stir until the caramel is smooth and silky.
Dip the tops of the cooled doughnuts into the caramel, then place them on a wire rack to allow the coating to set.
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