"We can't make bulgogi without beef."
Translation:소고기 없이 불고기는 못 만들어요.
Same doubt. Make is a transitive verb. So, I don't understand why 불고기 is not the object of 만들다. The example sentences on Wiktionary also have ~를 markers: https://en.wiktionary.org/wiki/%EB%A7%8C%EB%93%A4%EB%8B%A4#Verb_2
It is the object of the sentence! 은 and 는 don't mark a grammatical function; they can be used with both subjects and objects. The purpose of 은 and 는 is to emphasize the topic of the sentence, or what the sentence is about. Still, the sentence with "불고기를" should definitely be accepted.
If it were 없는, it would describe the 불고기: 소고기가 없는 불고기 "bulgogi without beef, bulgogi which doesn't have beef [in it]".
But that's not what is meant here - also because bulgogi which doesn't have beef in it is impossible because bulgogi is beef prepared in a particular way. Instead what we want to say is "without beef (under the circumstances of there not being beef) we cannot make bulgogi." So the "there is no beef" part does not describe the bulgogi but the verb action of making the bulgogi, which is why we need the adverb form 없이 rather than 없는 which would describe the following noun.
That said, 없게 can appear under certain circumstances, e.g. if the situation specifically calls for a -게 adverb (for example in the construction -게 하다 as 없게 하다 “to cause to be missing”). And certain idiomatic pairings of 없다 may also form the adverb in -게. For example 재수 없다 (a colloquial way of saying “to suck, to be really bad”) and 눈치없다 “tactless” can become either 재수 없게/눈치 없게 or 재수 없이/눈치 없이 – in fact the ones in -게 feel probably a bit more common to me. On the other hand 맛없다 usually becomes 맛없이 and 맛없게 is rarer.
There definitely seems to be a pattern when to use which suffix when both are possible, but it’s at times difficult to put into words. -게 seems to always be used, regardless of the verb, if the adverb tells you the result of the verb action (as in -게 하다, but it can also be used more flexibly, e.g. 커피를 맛없게 끓이다 “to brew coffee [in such a fashion that it becomes] ill-tasting”).
When we’re not talking about a result, things are more difficult. I feel like x없게 sounds more like a fashion in which the main verb is performed: It is done “x-lessly”. Whereas with x 없이 it feels more like a condition/state under which the main verb is performed. But take this with the appropriate grain of salt for feelings from a non-native speaker ;)
In any case 없이 is definitely by far the most common way to form an adverb from 없다.