Add the butter to a separate bowl and place a sieve over that bowl. Once the milk/egg mixture has thickened, pour it through the sieve over the butter and press it through with a spatula. You may notice a few small chunks of cooked egg in the sieve - this is totally okay! As my favourite Canadian pastry chef Anna Olson
always says, "Pastry chefs ALWAYS strain their sauces!" and for good reason!