Preheat the oven to 350°F (176°C) and adjust the oven rack to the center position. Line a baking sheet with parchment paper.
Peel, core, and slice each apple lengthwise into sixteenths. Place the slices in a single layer on the parchment-lined baking sheet. Bake the apples until they feel dry to the touch and take on a slight hint of color, about 20 minutes. Dump the apples into a bowl and let it cool. Leave the oven on.
In a bowl, combine the flour, sugar, baking powder, and salt.
Place the butter in a large bowl or the bowl of an electric mixer fitted with the paddle attachment. Add the apples, Cheddar cheese, cream, and 1 of the eggs. Sprinkle the flour mixture over the butter mixture and mix with a spoon or on low speed just until the dough comes together. Do not overmix.
Generously flour the work surface, place the scone dough on it, and sift a light layer of flour over the top. Use a rolling pin to gently roll the dough or use your hands to gently pat it into a 1 1/4 inch thick, 6-inch circle.
Cut the circle into 6 wedges (each wedge should be about 2 1/2 inches wide at its outer edge) and transfer the wedges to the baking sheet, leaving at least 2 inches between each scone.
In a small bowl, lightly beat the remaining egg with a pinch of salt. Brush the apple scones with the egg wash and sprinkle with the remaining 1 1/2 tablespoons of sugar.
Bake the wedges until firm and golden, about 30 minutes. With a spatula, lift the scones onto a wire rack to cool for maybe 10 minutes. The wedges are irresistibly fragrant and best when they're a little warm.