In the bowl of a stand mixer fitted with a dough hook (or in a large mixing bowl with a whisk), combine the water, granulated sugar, butter, egg yolks, yeast, salt, and vanilla extract on medium-low speed until combined.

In a medium bowl, whisk together the flour and baking powder, then gradually add to the yeast mixture until the dough is blended and pulling away from the sides of the bowl but still quite soft and sticky. Add a bit more flour, 1 tablespoon at a time, if necessary. (If blending by hand, use a wooden spoon until the dough becomes too thick, then switch to your hands. Knead for 10 minutes.)

Continue to knead on medium-low speed for 5 minutes, then transfer the dough to a large, oiled bowl. Cover with plastic wrap and let rest in a warm place until it doubles in size, about 1 hour.

Transfer the dough to a floured work surface and gently roll out to a 10x12-inch (25½x30½ -cm) rectangle. Trim any rounded edges if you wish, then cut the rectangle in half so you have two 5x12-inch (12½x30½-cm) strips. Cut each strip into 6 bars.

Place the doughnuts on 2 floured, parchment-lined baking sheets and cover with another baking sheet to keep them from drying out. (You can also use plastic wrap or a tea towel, but this dough is quite soft and may get stuck to whatever is touching it.)

Let rise in a warm place for about 1 hour, until doubled in size.

Line a baking sheet with a wire rack and place near the stove. Fill a large, heavy-bottomed saucepan with 2 inches (5 cm) oil, then clip in a candy thermometer and turn the heat to medium until the temperature reaches 350°F (180°C).

Fry the donuts, one at a time, for about 1 minute on each side, until golden. Let drain on the wire rack until completely cool.

In a shallow bowl, whisk together the powdered sugar, maple syrup, butter, and maple extract until smooth.

Dip the top half of each maple bar into the glaze, letting the excess drip back into the bowl before returning it to the wire rack to set.

Serve just after the glaze sets. These doughnuts are best just after they are made, but you can store leftovers in an airtight container at room temperature for up to 1 day.