Start by caramelizing your onion. Add your sliced onion to a preheated skillet with 1 Tbsp olive oil. Add salt and gently stir to coat all the slices.
Continue to cook on medium-low heat until the onion starts to break down and turn a golden brown color, about 40 minutes (see above photos for reference). Add more olive oil as necessary to prevent the onion from sticking to the pan.
Once the onion is golden brown and very soft, transfer to a bowl and set aside.
Add your sliced shiitake mushrooms to the pan with the dried thyme and/or rosemary if using. Add 1/4 cup of the mushroom broth or water and let the mushrooms cook, stirring occasionally, until most of the moisture has cooked off.
Place the cooked mushrooms in the bowl with the caramelized onion and set aside.
Add your chicken livers to the pan along with the chopped garlic. Season with salt and pepper, and add the remaining 1/4 cup of mushroom broth or water. Cook on medium-low heat until the liver is cooked through, but still remains pink in the center.
Transfer the liver, mushrooms and onion to your food processor and remove the food pusher from the feed tube (this helps heat escape while you're blending the patê). Pulse a few times to start, then blend until the mixture resembles a chunky paste.
Slice your butter into 1-2 Tbsp chunks.
Turn on your food processor and slowly feed the butter chunks through the feed tube while your patê is blending. By the time you've incorporated all the butter, your patê should be smooth and creamy, almost like mousse!
Transfer the patê to your desired storage containers and press a piece of plastic wrap on top of the surface to keep out any air. When stored like this, it will keep for 5-7 days in the fridge, or you can freeze it for up to 3 months.