Preheat the oven to 350°F. Line baking sheets with parchment paper.
Add the butter to a light-colored skillet or saucepan over medium heat. Cook, stirring or swirling occasionally, until the butter becomes deep golden amber and smells nutty. Remove from the heat, keeping all of the browned bits in the pan. Carefully stir in 1 tablespoon of water and let the butter cool slightly.
Add the chopped pecans to the same skillet and toast over medium-low heat for 3 to 5 minutes, stirring frequently. Remove from the skillet and let cool.
In a large bowl, combine the slightly cooled brown butter, brown sugar, and granulated sugar. Mix in the maple syrup, egg, egg yolk, vanilla extract, and maple extract until well combined.
In a separate bowl, whisk together the flour, old-fashioned oats, baking soda, cinnamon, nutmeg, cloves, and salt.
Gradually stir the dry ingredients into the wet ingredients just until combined. Fold in the toasted pecans. The dough should be thick and scoopable.
Scoop about 2 tablespoons of dough per cookie onto the prepared baking sheets, leaving room between the cookies for spreading. The dough does not need to be chilled.
Bake for 12 to 13 minutes, or until the edges are set and lightly golden and the centers still look soft.
Let the cookies cool on the baking sheet for 5 to 10 minutes before transferring them to a wire rack. Cool completely before glazing.
To make the maple glaze, whisk together the powdered sugar, maple syrup, milk, maple extract, and salt until smooth. If needed, add a very small amount of additional milk until the glaze is thick but drizzleable.
Drizzle the maple glaze over the completely cooled cookies. Leave the cookies uncovered for 1 to 2 hours, or until the glaze has set, before stacking or storing.