Grease a 2 1/2- to 3-quart baking dish. Heat oven to 375°.
Toss sliced apples with cranberries and lemon juice. In a bowl, combine the 3/4 cup granulated sugar, 1/2 cup brown sugar, 3 tablespoons flour, 1 teaspoon cinnamon, and 1/2 teaspoon salt; add to the apple mixture and toss to coat. Mix in the apple juice and vanilla. Arrange the apple mixture in the prepared baking dish. Dot with the soft butter. In a mixing bowl, combine 3/4 cup flour with 1/2 cup sugar, 1 teaspoon baking powder, and 1/4 teaspoon of salt. Whisk the egg with cream and melted butter; stir into the dry ingredients and beat until smooth. Using a tablespoon, drop spoonfuls of the batter evenly over the fruit mixture. The batter will spread as the cobbler bakes.
Bake for 40 to 50 minutes, until fruit is tender and topping is nicely browned.