Melt the 2 tbsp unsalted butter in a small saucepan over medium heat and cook, swirling occasionally, for 3 to 4 minutes until the solids turn amber and smell nutty. Add the 1/2 cup water carefully, then stir in the 1/2 cup granulated sugar and 1/8 tsp fine salt. Cook over medium heat for about 2 minutes until the sugar fully dissolves.

Bring the syrup to a gentle simmer and cook for 3 minutes without stirring. Remove from heat and stir in the 1/2 tsp pecan extract and 1/4 tsp vanilla extract. Cool to room temperature, at least 15 minutes.

Fill 2 tall glasses each with 1/2 cup ice, then add 1 cup chilled coffee per glass. Stir 3 tbsp of the butter pecan syrup into each glass for about 10 seconds.

Pour 1/4 cup whole milk into each glass, stir or layer as preferred, and serve immediately with a pinch of toasted chopped pecans on top if you have them.