Heat the milk, cloves and cinnamon sticks in a large pot over medium-low. Watch the milk closely, a layer of skin will form on top as it starts to simmer.

Once the milk simmers, lower and add the lime juice. It will curdle immediately. Stir until the milk simmers again then turn the burner off.

In a large, heavy bottomed pan (or caldero), heat the sugar over medium-low. The sugar will clump as it heats. Stir until it dissolves, allowing a few seconds between strokes. Turn the flame down low as the sugar starts to get dark. Once completely dissolved, proceed quickly to the next step.

Add a few ladlefuls of milk at a time to the pot with the melted sugar. Keep stirring as you add the milk and transfer the cloves and cinnamon too. The sugar will react by bubbling and hardening as milk is added. Keep stirring, adding a few ladlefuls of milk at a time. The mixture will stop bubbling and the sugar will dissolve completely as you keep stirring.

Turn the heat up slightly and let the mixture cook uncovered for 1 hour. Stir every 5 minutes or so, until two-thirds of the liquid has evaporated. As the liquid reduces, stir more frequently so it doesn't burn.

Once the mixture is reduced to two-thirds, stir in the vanilla extract and let cook a few minutes longer. Once you're satisfied with the texture and amount of liquid, turn the burner off. For thicker versions leave at least 1-2 inches of liquid so it doesn't dry out.