Preheat the oven to 350 degrees F. Line baking sheets with parchment paper.
How to make browned butter with pecans
In a medium saucepan, stir the butter (cut into small pieces) over medium heat until melted, stirring constantly. Continue to cook the butter over medium heat, simmering, stirring often. The butter will soon start to bubble and foam.
As the butter foams, keep stirring and reduce heat if needed. Keep a close look at the butter and as soon as it starts to turn brown (which you will see through the foam), add chopped pecans. The mixture will foam even more.
Continue cooking the butter with chopped pecans, constantly stirring, for not more than 1 minute, on low heat. Make sure to watch closely not to burn chopped pecans. Remove from heat and immediately pour this butter-pecan mixture into a separate, cool heatproof bowl.
DO NOT LEAVE the butter with chopped pecans in the same pan you cooked them, otherwise butter and pecans will burn! Cool the butter-pecan mixture in a separate heat-proof bowl for about 20-30 minutes, or even longer, depending on your kitchen temperature. The butter-pecan mixture should thicken and solidify.
How to make cookie dough for pecan sandies
Add powdered sugar, brown sugar, 3 tablespoons half-and-half, vanilla, and salt into the cooled butter-pecan mixture and stir until sugars are smoothly incorporated.
Mix in the flour until dough forms. If mixture is too crumbly and doesn't hold together, add small amounts of additional half-and-half, and continue to work the mixture. The dough should look like on my photo above (scroll up and you'll see step-by-step photos).
How to shape pecan sandies
Divide dough into 4 equal portions. Divide each ¼th of the dough into 9 equal balls. Roll each ball in coarse sugar. Place balls about 2 ½ inches apart on parchment-lined baking sheets; press a pecan half firmly into each ball. If edges of the cookies crack, just push back together.
How to bake pecan sandies
Bake one baking sheet at a time for 13 minutes until cookies start to brown.
Transfer baking sheet to a wire rack. Let cool 10 minutes for cookies to firm up. Using a wide spatula, transfer cookies to wire rack.
Optional: Brush pecans on top with maple syrup. Do it only to a few cookies, try them, and brush more cookies, if desired.