Place the butter and sesame seeds in a small saucepan over medium-low heat on the stovetop.
Cook the butter, swirling it often, until it turns deep amber in colour, about 12 to 14 minutes. Set the brown butter aside and let it cool for 10 minutes.
While the butter is cooling, preheat the oven to 375ºF (190ºC).
Line a standard 17-¼ x 12-¼ inch (44 x 31-cm) half sheet baking sheet with parchment paper.
In a medium-small bowl, whisk together the flour, baking powder, and salt
Pour the brown butter and sesame seeds into a large bowl. Add the brown sugar and whisk until smooth. Add the egg, and again, whisk smooth.
Dump half of the flour into the wet ingredients and gently fold until no streaks of flour remain. Add the remaining flour and again, gently fold until well combined.
Divide the cookie dough into 12 equal portions (approximately 35 grams each). Using your hands, roll each portion into a smooth ball.
Place the cookie dough balls on the prepared baking sheet.
Dip the base of a flat-bottomed drinking glass or mug into flour and use it to gently press each cookie ball until the edges of the dough start to crack.
Though floured, you will probably still have to gently peel the pressed cookie from the bottom of the glass/mug.
Place the sheet of flattened cookies in the preheated oven and bake them for 13 to 15 minutes, or until the edges and bottoms are golden. Cool the cookies on a wire rack completely before decorating.
To decorate the cookies, place the sesame seeds in a small, dry pan or pot on the stovetop.
Cook the seeds over medium-high heat, swirling often, until the seeds are fragrant and turn medium-dark gold in colour.
Spoon the seeds into a small heat-safe vessel and set aside.
Melt the chopped white chocolate in 30 second bursts in the microwave, or over low-medium heat in a small pot on the stovetop, until no lumps remain.
Dip the edge of each cookie in melted white chocolate then sprinkle with toasted sesame seeds.
Place the dipped cookies on a wire rack on the counter, or in the fridge, until the chocolate hardens, before serving or packaging as gifts.