In a large Dutch oven or large heavy-bottomed pot, heat 2 tablespoons butter until melted, then add cinnamon stick and coriander seeds, and toast, stirring constantly, until just fragrant, about 1 minute. Add onion, jalapeño, garlic, ginger, and salt and cook, stirring occasionally, until the onion just softens, about 2 minutes. Add sugar, stir, and continue to cook, uncovered, until sugar is dissolved, about 30 seconds.

Remove pot from heat and let cool slightly for 2 minutes. Stir in the apple cider, apple cider vinegar, Angostura bitters, if using, and mustard seeds. Return the pot to medium-high heat, and bring to a boil. Once boiling, stir in apples, then lower and adjust heat as needed to maintain a simmer. Cover pot and simmer until apples are tender and liquid has thickened enough that the base of the pot is dryish when you pull a spoon across the bottom, 30 to 35 minutes. (Test the apples texture by pressing apple pieces against the side of the pot with a spoon or spatula; when fully cooked, the apples will turn mushy and fall apart easily.)

Taste applesauce and adjust the flavor to taste by adding brown sugar for more sweetness; vinegar for more acidity; and/or jalapeño for more heat, if needed. If adding ingredients to taste, continue to cook for 1 additional minute after the additions, then remove from heat. Discard the cinnamon stick.

Depending on your preference for consistency, mash the applesauce with a potato masher, blend with an immersion blender, or purée part (or all) in small batches in a blender or food processor.

Return mashed or puréed applesauce to the pot, cover, and bring to a simmer over medium-low heat, stirring frequently until heated through and the flavors meld, about 10 minutes.

Remove from heat, stir in lemon juice and remaining 1 tablespoon butter. Serve warm, at room temperature, or cold.