Place racks in upper and lower thirds of oven and preheat to 350°. Pulse 9 graham cracker sheets (about 5 oz.; 1 sleeve) in a food processor until sandy crumbs form. Add ¾ cup plus 1 Tbsp. (101 g) all-purpose flour and ¾ tsp. kosher salt; pulse to combine. Transfer to a small bowl.
Pulse ½ cup light brown sugar and ½ cup (1 stick) unsalted butter, room temperature, in processor until creamy. Add 1 large egg and ½ tsp. vanilla extract and pulse just to combine. Add graham cracker mixture and process until dough comes together and forms a ball, about 45 seconds.
Place ¼ cup granulated sugar in a clean small bowl. Using a #40 cookie scoop (about 2 Tbsp.), portion out dough; roll into smooth balls (dough will be sticky). Working one at a time, toss balls in granulated sugar to coat generously. Divide between 2 parchment-lined baking sheets, spacing at least 2" apart (about 7 per sheet). Working one at a time, dip the bottom of a shot glass or ⅛-cup measuring cup into remaining granulated sugar, then press down in center of balls to create a divot that can hold 2 tsp. filling, about ¼" deep. Chill cookies in freezer 15 minutes
While cookies are chilling, whisk 1 large egg yolk and ⅓ cup sweetened condensed milk in a small bowl to combine. Add 2 Tbsp. plus 1½ tsp. fresh lime juice and 4 tsp. finely grated lime zest; whisk until slightly thickened.
Remove cookies from freezer and spoon about 2 tsp. filling into each divot (don’t overfill). Bake cookies, rotating baking sheets top to bottom and front to back halfway through, until filling is set and cookies are firm and golden brown around the edges, 12–14 minutes. Let cookies cool on baking sheets 5 minutes. Transfer to a wire rack and let cool completely
Just before serving, finely grate a little lime or lemon zest over each cookie.
