Heat the oven to 375°F. Put the baking mix in a bowl. Add the egg to the cream and stir with a fork to combine; add the sugar and stir. Make a shallow well in the dry mix and pour in the wet ingredients. With as few strokes as possible, stir until the mixture is just combined. It should be loose, but if it doesn’t hold together at all, add more cream 1 Tbs. at a time.
Generously flour a counter and turn the dough out onto it; it will be shaggy and very soft. Scrape any bits in the bowl onto the mound of dough. Flour your hands. Shape the mixture into a very loose rectangle with the short side nearest you. Using a bench knife, a pastry scraper, or a spatula to help lift the ragged dough, fold the bottom third over the center third, and then the top third over the center, as if you’re folding a business letter. Pat down to shape another rectangle, turn it so the short side is nearest you, and repeat the folding. Pat down again to about 1/2 inch for a cobbler, 3/4 inch thick for shortcakes.
For a cobbler—Toss your fruit with the sugar and flour and spread in a 9x13-inch baking dish. Lay the dough on top. It’s fine if the topping breaks into pieces; just be sure most of the fruit is covered. Brush with cream and sprinkle with sugar. Bake until the topping is golden brown, 25 to 30 minutes.
For shortcakes—Cut the dough into the shapes you want—circles or squares—and set them on an ungreased baking sheet (or one lined with parchment). Brush with cream and sprinkle with sugar. Bake until the tops have started to brown and the bottoms are golden, 20 to 22 minutes. To make a filling for shortcakes, slice the strawberries (if using), and toss all of the berries with 1 Tbs. sugar, adjust the sweetness to taste with the lemon juice and remaining sugar. While the biscuits are still slightly warm, split them and fill with the berries, and top with whipped cream.
Try flavoring the shortcakes or cobbler topping to suit the fruit: a pinch of cinnamon and ginger into the dry mix for an apple cobbler; 1/2 cup chopped nuts to the dry mix for a peach shortcake. Extracts and zests are another option: along with the cream, try adding 1 tsp. vanilla extract for strawberry shortcakes or 1/2 tsp. almond extract for an apricot or plum cobbler.