Preheat oven to 450 degrees.
In colander under cold running water, scrub sweet potatoes. Prick the potato skins in several places. Plop the whole potatoes on a sheet of heavy-duty aluminum foil or a baking sheet. Bake in preheated 450 degree oven until potatoes are tender. Cooking times will vary according to the size of the potatoes. Most take 20-30 minutes. A good way to test for doneness is to probe the potatoes with a skewer.
Meanwhile, put the apples and cider in a small saucepan. Simmer over low heat until most of the liquid has been absorbed, so the pan is almost dry, 10-15 minutes. Remove from the heat.
Peel the sweet potatoes, holding them in a towel to protect your hands from the heat. Drop them into a pot or large bowl. Smash the flesh with a potato masher or a heavy wire whisk, then whisk until smooth. Or run them through a potato ricer and then mash. Whisk in the butter. Stir in the apples and lemon juice. Season with cayenne and salt. Serve warm.