In a large bowl, sprinkle the yeast over the water and add a pinch of sugar; set aside for 5 minutes, until it gets foamy. Add the remaining sugar, flour, butter, egg and salt and stir until the dough comes together; knead (I use the dough hook of my stand mixer) for 5 minutes, or until the dough is soft and elastic, but still tacky. Cover with a tea towel and set aside for an hour.
Meanwhile, cook the butter, apples, sugar and a shake of cinnamon for 3-5 minutes, until the apples are soft and not too liquidy. Set aside to cool completely.
Roll or pat the dough out into a large rectangle (about 8x12-inches) and spread half with the cooled apples. Fold the other half over to cover the apples, pinch to seal and roll again until it’s about as thin as it originally was. (Don’t worry about the apple pieces poking through.) Fold it over itself and roll again, then cut with a knife or bench scraper into 1/2-inch strips in both directions — essentially you’re cutting the apple-filled dough into 1/2-inch pieces.
Flour the board or countertop and gather the dough and apples into balls, and shape into patties about the size of your palm, sprinkling with flour as they get squishy and the apples try to escape. They’ll look messy and rough — that’s OK! Put them on a parchment-lined sheet, cover with a tea towel and let rise for half an hour. To make the glaze, whisk together the icing sugar, maple syrup and water.
When you’re ready to cook your fritters, heat a couple inches of oil in a deep skillet or shallow pot until it’s hot, but not smoking — if you have a thermometer, it should read about 350?F. Cook the fritters a few at a time, without crowding the pot, flipping with a slotted spoon as they turn deep golden. Transfer to a paper towel-lined plate to absorb any excess oil, and drizzle generously with glaze while they’re still hot. Makes about a dozen fritters.