I want to make compound butter using my powerful blender. I am having trouble doing this with solid or soft butter because the blender cannot create a vortex and mix as well compared to blending with liquids. I was thinking of melting the butter and adding ingredients (e.g., shallots) and blending it into a beautiful liquid, and then refrigerating it. What would this do?
You can do this, but it won't have the same texture when it resolidifies. It'll likely be softer and more translucent. See this answer for a bit about why this happens.
I make lazy garlic butter sometimes by melting and gently heating the garlic in it (not enough to fully cook it, just to soften the garlic and mellow and release the flavor). It definitely works, in terms of producing something that tastes like butter and garlic.
I don't know if you necessarily need the blender for this. Mincing is plenty to make the flavors release well, and soft pieces of minced garlic don't bother me. I believe it'd work if you blend while melted if you want. I'd probably try not to overdo it, though, because you're likely going to be breaking the remaining fat globules and disrupting the texture even more.
One compromise might be to blend with as small an amount of melted butter as necessary to get your blender to work well, and leave the rest of the butter soft and unmelted to mix into.
Of course, blenders just aren't that well-suited for this task. A mini food processor is a much, much better way to chop small amounts of solid things, so if you don't mind another smallish purchase, that'd be my first choice.