Baking – Browned butter + roasted flour = baking magic

Your recipe is suspect. Melted butter is the same in chemical composition as softened butter. The two react a bit differently in recipes due to the way that they interact with the other ingredients to impart textures: in short, the creamed butter will hold microscopic bubbles that the fully melted butter will not. Adding egg yolks will help the melted butter incorporate, and probably lead to more cakey cookies (compensating for the flatness you get from melted butter), but the effect is hard to isolate.

As for brown butter, what happens is that as moisture evaporates from the butter, the boiling point increases (similar to what happens when making caramel or candy). As the boiling point of the butterfat increases, the temperature of the butter solids (which consist of lactose and milk proteins) begin to caramelize (the sugars) and undergo a maillard reaction (the protein). Both of these reactions will impart unique flavors to the butter, yielding the distinctive brown butter taste.

As the butter heats, the integrity of the fat is damaged. This occurs to all fats, which is why you can only use a batch of fat for deep frying a few times before it needs to be discarded.

Dark chocolate is actually more heat resistant than other kinds. Milk chocolate can scorch at 115° F / 46° C or higher; semi-sweet can have problems at 125° F / 52° C or so. It's mainly the dark chocolate that can withstand temperatures as high as 140° F / 60° C.

I think there are two misconceptions to clear up here, those being:

• Baking at 350° F does not mean you are raising the internal temperature to anywhere even close to 350° F. Even yeast breads are not baked to an internal temperature higher than around 200° F, and it's much lower for soft breads and especially cookies. It's hard to find much data on the recommended internal temperature for cookies, but I'd estimate it to be around 160° F simply for food safety reasons.

• Just because a food can burn above a certain temperature, does not mean that it will happen immediately. Just as heating oil slightly above its smoke point does not immediately result in flames and rancid taste, heating chocolate above the aforementioned temperatures will not immediately cause it to scorch. What it means is that chocolate can be sustained in a melted state indefinitely at lower temperatures, but raising the temperature further will cause it to eventually scorch. The higher the temperature, the less time it will take.

These "scorching" temperatures are mainly cautions against direct stovetop or microwave heating; with these methods it is easy to get the temperature very high, very fast. When you bake cookies, you are applying very slow, gradual heat; if the temperature of the chips even gets as high as 140° F, it won't stay that high for very long. It's not enough to burn, and that is why even milk chocolate or white chocolate chips tend to do fine in cookies.

So just bake them, and don't worry. The cookies themselves will probably scorch before the chocolate does.

P.S. As SAJ14SAJ writes in the comments, this assumes that you bake the cookies on some kind of insulating material like parchment or a baking mat. There is a risk of burning if you bake them directly on a metal tray and the chocolate comes into direct contact with it, as it will then heat up very quickly to approximately the same temperature as the tray. So don't do that - if you use a metal tray or cookie sheet, make sure there's something between that and your cookies. They bake from the ambient heat in the oven, and don't really need direct heat from the tray.