Peel and core the apples. (A corer is helpful for this task, but you may also quarter the apples with a large knife and then cut out the core.)
Cut apples into quarters, then cut into large cubes.
Heat sugar in a pot over medium heat until light brown in color, stirring constantly.
Add cinnamon stick and allspice. Pour apple juice into pot, stirring vigorously, until sugar is dissolved.
Add apples to pot, cover and cook over low heat, stirring occasionally, until softened, about 12 minutes. Transfer apple compote to a bowl and let cool completely. Serve compote over pancakes, waffles, rice pudding, or cereal, as desired.