Combine 1/2 cup packed brown sugar and 1/4 cup water in a small saucepan over medium heat; stir constantly until the sugar dissolves, about 2 minutes.

Remove the saucepan from heat and let the brown sugar syrup cool to room temperature, about 5 minutes.

Brew 2 ounces of espresso or strong coffee into a coffee mug or glass; if using instant coffee, dissolve 1 teaspoon of instant coffee in 2 ounces of hot water.

Pour 2 tablespoons of cooled brown sugar syrup into the hot coffee and stir until fully combined.

Pour 8 ounces of milk into a small saucepan and heat over medium heat until small bubbles form around the edges, about 3 minutes.

If desired, froth the heated milk using a milk frother or whisk for about 30 seconds to create a light foam.

Slowly pour the hot milk into the coffee and syrup mixture, holding back foam with a spoon; pour in a thin stream first, then more generously to fill the cup.

Spoon a thin layer of milk foam on top of the latte, about 1/4 inch thick, for a café-quality finish.

Stir in 1/4 teaspoon of vanilla extract if using, then serve immediately.