Looking for a pistachio cake recipe, very moist, maybe a syrup over it, like the one at Le Pain Quotidienne

I've made this cake with pistachios instead of almonds, and it come out similar to your description (and is excellent): http://www.lottieanddoof.com/2012/05/almond-olive-oil-cake/

Ok, this one is a lot of effort, but in my opinion, so worth it: Milk Bar Pistachio Cake
https://trissalicious.com/2012/03/21/facing-fear-milk-bar-pistachio-cake/
http://flourtrader.blogspot.com/2013/03/pistachio-layer-cake.html

I listed 2 recipes because one is in metric and the other is in US-friendly volume (?).

A former co-worker made this, twice, and it was amazingly delicious each time. While not difficult, it is a cake that required multiple elements that you could make ahead. I can vouch that the final product is worth the effort. There are good reasons it is a best seller for Milkbar.

See this pistachio tart, which includes some brown sugar and corn syrup in its filling. Haven't made the recipe but it looks similar.
https://www.realsimple.com/food-recipes/browse-all-recipes/pistachio-tart

https://smittenkitchen.com/2017/04/pistachio-cake/

The fig and pistachio cake at the end of this Mimi T post is super easy to make, beautiful, and tasty http://mimithorisson.com/2016/08/02/2-recipes-from-my-new-cookbook/

Her walnut cake is also great! https://food52.com/recipes/64753-walnut-cake