Melt butter in a small saucepan over low heat. Remove from heat and stir in lemon juice.
Peel apples nearly halfway, leaving skin on bottom halves. Carefully cut out apple cores with a paring knife, leaving bottoms intact. Holding the paring knife at a 45 degrees angle, widen the opening at the top of each apple by about ½ inch. Place apples in an 8-inch square or round baking dish. Brush peeled parts of apples with 1 tablespoon butter mixture and sprinkle with 1 tablespoon sugar.
Pulse granola in a food processor until it forms coarse crumbs. Add egg white, remaining butter mixture and 1 tablespoon sugar; pulse until just combined.
Spoon the granola mixture into apples, spreading it slightly over tops. Sprinkle with remaining 1 tablespoon sugar. Pour cider into baking dish around apples.
Bake the apples until tender and filling is puffed and lightly browned, 50 to 60 minutes. (Cover apples loosely with foil if browning too quickly.) Cool in the pan for about 30 minutes. Serve warm, spooning pan juices over apples.