The compound butter is basically just butter that's had herbs/flavors added to it. You roll it up into a log... and when the time comes, you slice some off and melt it on top of a steak or chicken, or whatever.... to give it a nice boost of flavor.
A beurre manie is a mixture of butter and flour (mine will be gluten free, naturally) and you use it to thicken sauces without the necessity of creating a roux. You mix the butter, flavorings, and flour.... and then drop bits in to the sauces. They thicken and flavor. =)
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