Rinse the apples and orange thoroughly: Quarter the apples without peeling or coring them. Quarter the orange as well. Place all of the fruit in an 8-quart stockpot.
Add the cinnamon sticks, whole cloves, and allspice berries. Pour in 10 cups (2.4 L) water, or enough to cover the fruit by about 1 inch. The fruit may float initially, which is normal.
Set the pot over high heat and bring the mixture to a boil. Once boiling, reduce the heat to low or medium-low so the liquid maintains a gentle simmer rather than a vigorous boil.
Cover the pot partially and simmer for 1½ hours. Stir occasionally and press floating fruit beneath the liquid. By the end of this stage, the apples should be very soft and beginning to break apart.
Use a potato masher or sturdy wooden spoon to mash the softened apples and orange directly in the pot. Break the fruit into a coarse pulp rather than trying to puree it completely. This exposes more of the fruit to the liquid and strengthens the cider flavor.
Continue simmering the mashed fruit uncovered for another 30–45 minutes. The liquid should become deeper in color and strongly aromatic. If the level drops dramatically, reduce the heat rather than adding a large amount of extra water.
Remove the pot from the heat: Place a fine-mesh strainer over a large heatproof bowl or clean pot. Carefully ladle the cider and fruit into the strainer, pressing gently on the solids to extract the liquid. For a less pulpy cider, line the strainer with cheesecloth and strain a second time. Discard the cooked fruit and whole spices.
While the strained cider is still hot, stir in the brown sugar and vanilla extract. Taste carefully. If your apples were especially tart, add additional brown sugar 1 tablespoon at a time until the sweetness is balanced. The cider should taste apple-forward rather than syrupy.
If the cider tastes too concentrated, stir in a small amount of hot water, about ¼ cup (60 mL) at a time. If it tastes too weak, return it to the pot and simmer uncovered for several minutes to reduce slightly.
Let the cider stand for about 30 minutes if you plan to refrigerate it. For immediate serving, it can be ladled into mugs while still warm. Do not seal very hot cider in an airtight storage container.
Serve warm in mugs or chill completely for cold cider. If desired, garnish individual servings with an optional cinnamon stick, apple slice, or orange slice.