In a medium saucepan, combine the ketchup, brown sugar, pineapple juice, Worcestershire sauce, and apple cider vinegar. Stir the mixture over medium heat until the sugar dissolves completely.
To the saucepan, add the chopped scallions, minced garlic, scotch bonnet peppers, and grated ginger. Stir in the ground allspice, kosher salt, dried thyme, ground cinnamon, and ground nutmeg.
Bring the mixture to a gentle simmer. Allow it to cook for about 30 minutes, stirring occasionally, until the sauce thickens to your desired consistency.
For a smoother texture, transfer the sauce to a blender or use an immersion blender directly in the saucepan. Blend until smooth.
Let the sauce cool to room temperature. Transfer it to an airtight container and refrigerate. The sauce can be stored in the refrigerator for up to two weeks.