Put ¾ cup sugar, ½ cup unsweetened cocoa powder, ½ cup milk, and a 1 teaspoon vanilla extract into a medium saucepan. Whisk them all immediately to begin combining. The mixture will look grainy and lumpy.
Set the pan over medium-low heat and whisk it constantly as the mixture warms. The sauce will gradually change from a thick, lumpy paste into a smooth, glossy liquid as the sugar dissolves and the cocoa incorporates, about 4–5 minutes.
Continue whisking until you see small bubbles forming around the edges of the pan. Do not allow the sauce to come to a rolling boil. Once you see these first bubbles, immediately remove the pan from the heat.
Turn off the heat, whisk in 1 teaspoon vanilla extract until it is fully incorporated.
Let the sauce cool in the pan for 5 to 10 minutes, whisking occasionally to prevent a skin from forming.
Transfer the sauce to a clean glass jar or airtight container. Once it's cooled to room temperature, refrigerate it for up to 2 weeks.