In bowl of stand mixer (or in large bowl), dissolve sugar in milk; sprinkle yeast over milk mixture and let stand until frothy, about 10 minutes.

With paddle on low speed (or with wooden spoon), beat in butter, egg yolk, salt and half of the flour. Beat in remaining flour, 3 tbsp/45 mL at a time, until dough is soft and sticky but just holding together. Scrape into buttered bowl; cover with plastic wrap. Refrigerate for at least 4 hours or, preferably, overnight.

Caramel-Apple Filling: Peel, core and cut apples into 3/4-inch/2 cm cubes; toss with lemon juice and set aside.

Stir sugar with 3 tbsp/45 mL water in small heavy-bottomed saucepan; place over medium heat and cook, without stirring, until sugar is caramel-coloured. Stir in apples and any juice; cook until tender, 3 to 5 minutes.

Into second saucepan, strain cooking liquid from apples; transfer apples to bowl. Return cooking liquid to pan and cook until thick enough to thickly coat spoon and reduced to about 1/3 cup/75 mL. Remove from heat; stir in butter until thoroughly combined. Stir into apples to coat; let cool.

On lightly floured surface, roll out dough into 14-inch/35 cm square. Cut dough in half, then cut each half into 4 pieces to make 8 rectangles in total. Spoon 1/4 cup/60 mL of the filling onto centre of one half of each rectangle; brush edge with egg white. One rectangle at a time, fold dough over filling to make beignet; firmly press edges to seal.

Into deep fryer or deep saucepan, pour enough oil to come 2 inches/5 cm up side; heat to 360F/185C. Cook beignets, two at a time and turning often, until golden brown, 1-1/2 to 2 minutes.

Using slotted spoon, transfer to paper towel–lined rack; let cool. Mix icing sugar with cinnamon; sprinkle through fine sieve over tops.

Makes 8 beignets.