Ah, darn, Peachey beat me to it! Teacher knows best :) I'll just try to rephrase it for you. You can have "best" as an adverb, describing a verb: he runs best, she works best… "Best" can also be an adjective, with a noun: he's the best runner, she's the best worker… When using it as an adjective, when it's describing a noun, you should use the article "the". That is the most common function you are used to seeing it in and that's why you feel the need to add the article. Just consider what it's referring to and the answer is easy.

I believe it is because there is no subject/noun in the first sentence. There is nowhere to use the article. In the second sentence you are using 'the' to denote the noun 'choice'. In the first sentence there is no noun. However, you are correct about it being a set phrase also. Sometimes sayings like these don't necessarily follow the rules.

Superlatives are usually taught as adjectives (which is why students are used to seeing "the" in superlative phrases), but superlatives can also work as adverbs, as in "what works best". The superlative describes the verb, and not a noun (which needs "the"), so as far as regular grammar goes, it's perfectly correct.