Set a rack in the top third of the oven and preheat to 400F.
Line 2 cookie sheets with parchment and set aside.
Put the sesame seeds in a dry skillet and heat over medium heat, stirring occasionally and then constantly as the sesame seeds begin to take on a bit of color. Toast the sesame seeds to a light golden brown. Spread out on a paper towel to cool.
Wipe out the skillet in case any sesame seeds are stuck to the bottom (just to be safe) and fill with about 1/4" vegetable oil. Allow the oil to get hot. Test by dropping in a piece of slivered almond. It should begin to gently bubble in 1-2 seconds.
Add the rest of the almonds and fry, stirring constantly, until a very pale golden color. The nuts will continue to take on color for an additional couple of minutes, so be sure to remove them from the oil before you think they're ready.
Pour the nuts into a strainer set over a heat-safe bowl and then spread them out to cool completely.
Coarsely grind the nuts in a small food processor or mini chopper. Set aside.
In the bowl of a stand mixer, combine the granulated sugar, vanilla sugar (or extra granulated sugar plus 1/2 teaspoon vanilla extract), butter, oils, and salt.
Beat until reasonably well combined (it may look a bit lumpy. That's okay).
Mix in the sesame seeds and ground almonds.
Whisk together about half the flour and baking powder, and then add that to the mixer. Mix on low speed until you have a sticky dough.
Add enough of the remaining flour to make a stiff, crumbly dough that holds together when you squeeze it. I ended up using 450 grams of the total of 500 grams of flour. You may need a bit more or less depending on your flour and the humidity.
Once the dough is the right consistency (and even if it doesn't crack, it will be delicious, so don't get upset), weigh it and divide the weight by 40. I ended up needed 22 3/4 grams of dough per cookie, so some I scaled at 22 grams and some at 23 grams and it all worked out just fine.
Form each piece of dough into a fairly smooth ball (you won't be able to roll it, because it will crumble, so just form it with your hands). Flatten each ball of dough between your palms so the edges are thinner than the centers. Try to make them as round as you can, but it's not the end of the world if they're lopsided.
Place each shaped cookie on the prepared sheets--20 per sheet.
Bake one tray at a time for 7 minutes. Then rotate the pan 180 degrees, reduce the heat to 375F and bake an additional 3-6 minutes (depending on your oven) until very pale golden around the edges. With all the seed and nut oils, they'll take on color quickly, so keep an eye on them.
Remove the sheet from the oven and carefully transfer to a cooling rack. The cookies will be soft, but they will crisp and crunch up as they cool. By the time they are completely cool, they'll be perfect!
Bake the second sheet (remember to turn the oven back up to 400 first). The second batch might take slightly less time to bake than the first batch. Mine did.
Try not to eat all of them. Enjoy!