In 1-quart saucepan, melt butter over low heat. Stir in sugar until it dissolves, about 20 seconds. Add apples and shallot. Cook 20 to 30 minutes, stirring occasionally, until mixture turns golden brown and aroma is reminiscent of apple cider. (Do not rush process; color takes a while to form, but the flavor is worth it.) Set aside to cool completely.