In a large, microwave-safe mixing bowl, melt the butter. Starting with a stick from the fridge, I will typically microwave the butter for around 30 seconds followed by 10 second intervals until the butter is mostly melted, then stir it until it is completely melted.
Add the brown sugar to the butter and whisk together until the mixture is pretty uniform.
Prepare the Egg Replacer according to the package directions. Then mix the Egg Replacer and vanilla extract into the butter mixture.
In a separate mixing bowl, stir together the oats, flour, baking soda and salt.
Pour the dry ingredients into the butter mixture. Then use an electric hand mixer on low speed to mix everything together just until all the dry ingredients are mixed in.
Use a spatula to scrape the sides of the bowl and finish mixing the dough. Then form the dough into a large ball, cover with plastic wrap and refrigerate for at least an hour up to 3 days.
Preheat the oven to 350°F. While the oven preheats, remove the cookie dough from the fridge and let it warm until it is just soft enough to work with.
Form the cookie dough into golf ball-sized balls and arrange them on a cookie sheet about 2 inches apart.
Bake on the center rack for 10-12 minutes, just until the edges of the cookies start to turn golden brown.
Remove the cookies from the oven and allow them to cool on the baking sheet for at least 5 minutes. Then transfer the cookies to a cooling rack and cool for at least 5 more minutes before serving.
Per Serving: 163 calories; 8g fat; 22g carbohydrates; 1.8g protein.