In a large bowl, whisk together the all-purpose flour, rye flour, sugar, salt, and baking powder to combine. Set aside.

In a medium bowl, combine the yeast and lukewarm water and whisk thoroughly to dissolve the yeast. Add the milk and eggs and whisk thoroughly until smooth.

Make a well in the centre of the dry ingredients, then pour in the wet ingredients. Use a bowl scraper or a large flexible spatula to mix until all of the flour is hydrated and you have a very wet, evenly mixed dough that almost looks like a thick batter. Use the scraper or spatula to scrape down the sides, making sure there’s no unincorporated flour in the bottom of the bowl, then clean off the scraper and set it aside.

Hold the bowl steady with one hand and use the other to grasp a handful of dough from along the side of the bowl, then stretch it upward and fold it into the center. Rotate the bowl slightly, then repeat this motion, lifting a portion of the dough and pressing it back into the center. It will be very loose and sticky but will gradually firm up and start to hold its shape as you work. Continue to knead, pausing occasionally and using the bowl scraper or spatula to clean the dough from your hand and scrape down the sides of the bowl, until the dough is elastic and puts up some resistance as you lift it, 10 to 12 minutes (it will still be very sticky, though less sticky than before). Scrape down the sides of the bowl one more time and gather the dough into a single mass in the bottom.

Cover the bowl and let the dough sit at room temperature until it’s doubled in size and bubbly across the surface, 1 to 1 ½ hours. Transfer the covered bowl to the refrigerator and chill until the dough is cold, at least 4 hours (this will help develop the flavor of the beignets and make the dough easier to handle)

Clip a deep-fry thermometer to the side of a medium Dutch oven or large heavy-bottomed saucepan. Pour in the oil until it reaches between one-third and halfway up the sides, but no higher (if you have any remaining oil, save it for another use). Heat the oil over medium-high heat until it reaches 325°F. Reduce the heat to medium and continue to heat the oil until it registers 375°F on the thermometer. While it’s heating, line a large sheet pan with paper towels and set a wire rack on top, then place it next to the stovetop.

Very generously flour an 18 × 13-inch area of work surface, then remove the dough from the refrigerator and scrape it out onto the floured surface. Generously flour the surface of the dough, then pat it down with your palms into a rectangular slab measuring about ⅜ inch thick, 18 inches long, and 13 inches wide, adding more flour as needed to prevent sticking. Use a wheel cutter to cut the slab into 2 ½-inch squares. Discard any scraps from along the edges, then quickly separate the squares on the work surface and dust with more flour to prevent sticking along the cut sides. Cover the beignets with a clean towel.

When the oil is ready, uncover the dough, pick up a square, and toss it back and forth gently between your hands to dust off excess flour, then drop it carefully into the oil. Repeat with more squares of dough, adding only as many as can comfortably fit in the Dutch oven or saucepan with some room to swim around (keep the remaining pieces covered on the work surface). Fry the beignets, turning them several times with a slotted spoon or spider, until they’re puffed and deep golden brown all over, 2 to 3 minutes total. Use the spider to transfer the beignets to the wire rack, then repeat the frying process with the remaining squares of dough in additional batches, transferring the beignets to the wire rack as they’re done.

While the beignets are still hot and resting on the wire rack, dust them very generously with confectioners’ sugar. Transfer the beignets to a plate or platter and serve immediately.