Make the filling: In a large sauté pan, melt the butter over medium heat. Add the apples and sauté until just beginning to become tender, 3 to 4 minutes.
Add the lemon zest, lemon juice, brown sugar, and cinnamon, and sauté for 1 minute more.
Remove the mixture from the pot and cool to room temperature.
Make the fritters: Heat 3 to 4 inches of oil over medium-high heat until it reads around 350°F on a thermometer (or throw a scrap piece of dough in as a tester—if the dough bubbles and rises to the surface, the oil is good to go).
While the oil heats, roll out the doughnut dough on a lightly floured surface to 1/4 inch thick—no need to be precise in size or shape, as rustic is the name of the game here.
Scatter the cooled apple filling in an even layer over the dough. Roll the dough up into a log, then use a rolling pin to roll the log flat—no more than 1 inch thick.
Cut the dough into squares (about 3 x 3 inches). Let the fritters rise, covered, on a lightly floured surface for 30 minutes to 1 hour. When the oil is hot, gently add the doughnuts to the pot in batches, frying until they’re golden brown on both sides, 3 to 4 minutes. (Note: Some of the fruit may fall out—that’s okay! If you’re frying multiple kinds of doughnuts, fry these last, as escaped filling may color/flavor the oil.)
Drain the doughnuts on several layers of absorbent paper towels. Let cool for 5 minutes, then toss in cinnamon sugar to coat. Serve immediately.
I always carry three kinds of hot sauce in my purse. I have a soft spot for making people their favorite dessert, especially if it's pie. My first cookbook, The Fearless Baker, is out on October 24, 2017.