French Toast Pecan Cookies

Preheat oven to 375 degrees F (190 degrees C). Lightly grease baking sheets. Combine flour, white sugar, brown sugar, cinnamon, and baking soda together in a bowl, breaking up any large lumps of brown sugar. Stir butter, egg, vanilla extract, and rum-flavored extract into flour mixture until dough forms; add chopped pecans and mix until well blended. Roll dough into 1-inch balls and place 2 inches apart on prepared baking sheets. Press a pecan half on top of each ball. Bake in the preheated oven until edges of cookies are golden brown, 9 to 11 minutes. Sprinkle salt onto each cookie and gently press each with the back of a spatula to set the salt.