Brew the chai tea and set it aside. If making an iced latte, let it cool slightly; if making a warm latte, keep it hot.

In a small saucepan over low heat, combine 1/2 cup of the milk, 2 tablespoons pumpkin puree, 2 tablespoons caramel sauce, maple syrup or brown sugar if using, 1/4 teaspoon pumpkin pie spice, vanilla extract, and a pinch of salt.

Whisk the mixture until smooth and steaming, then remove from the heat. Do not let it boil.

Pour the chai tea into a mug or heatproof glass, then add the pumpkin-caramel milk mixture and the remaining 1/4 cup milk. Stir well.

For the pumpkin cold foam, combine the heavy cream, cold milk, 1 tablespoon pumpkin puree, powdered sugar or vanilla syrup, and 1/4 teaspoon pumpkin pie spice in a tall cup or jar.

Froth the foam with a handheld frother, immersion blender, or whisk until thick, creamy, and pourable.

If serving iced, fill a glass with ice before adding the latte. Spoon the pumpkin cold foam over the top.

Drizzle with additional caramel sauce and sprinkle with a little extra pumpkin pie spice. Add whipped cream if desired and serve immediately.