Preheat oven to 350. Cover a large baking sheet with parchment paper.
Whisk the flour, baking soda, and salt in a medium bowl.
Add the butter to a heavy skillet or pot. Cook for 6-9 minutes, stirring constantly. As it cooks, it will foam on the edges. It will turn golden brown, and the milk solids will be at the bottom of the pot. Remove the browned butter to a heat-resistant bowl to stop the browning process. Cool the butter in an ice bath just until it begins to resolidify.
Whisk the instant coffee with a tablespoon of hot water.
Using a handheld or stand mixer with the paddle attachment, on medium speed, cream the browned butter, brown sugar, and granulated sugar.
Reduce the speed to low and add the egg and vanilla extract, mixing just until incorporated. Beat in the flour mixture in a couple of increments, mixing just until combined.
Using a cookie scoop, drop the dough about 2½ inches apart on the prepared baking sheet. Bake the cookies for 10-11 minutes until very lightly browned and set.
In a saucepan over medium-low heat, melt the butter in the maple syrup, whisking constantly. Once the butter has melted, remove it from the heat and whisk in the powdered sugar, vanilla extract, and salt. Continue whisking the mixture until it is smooth and has a good drizzling consistency. Drizzle the glaze over the fully cooled cookies.