In a bowl, combine the warm water, sugar and vanilla. Set aside.
In a large bowl with a wooden spoon or in a stand mixer (using the dough hook), combine the flour, yeast, baking powder, and salt. Add the milk mixture and eggs and beat thoroughly until the dough begins to form. Gradually stir in the softened butter and knead the dough for about 5 minutes in the stand mixer or on a floured work surface. Place the dough in a clean and lightly oiled bowl. Let stand in a warm and humid place for about 2 hours or until the dough has doubled in size.
On a lightly floured surface, roll out the dough to a 1-cm (3/8-inch) thickness. With a 9-cm (3 ½-inch) and a 4-cm (1 ½-inch) cookie cutter, cut about 12 donuts and 12 donut holes, reusing scraps. Let stand for 15 minutes.
Preheat the oil in the deep fryer to 190 °C (375 °F). Line a baking sheet with paper towel or top with a rack.
Fry about four doughnuts at a time for about 1 minute per side or until golden brown. Drain on the baking sheet. Continue with the remaining dough. Let cool.
With a pastry bag fitted with a plain tip or a butter knife, frost the top of each donut with maple butter.
The doughnuts are better the same day.
Recipes with the "Healthy Pick" stamp have been evaluated by a registered dietitian member of the Ordre professionnel des diététistes du Québec.