The second part is not a dangling modifier.
Instead, the second part is an adverbial phrase that modifies smeared. It indicates how the tiny amount of edible oil was smeared.
"First, a shallow pan is smeared with a tiny amount of edible oil."
"How is this done?"
"[By] using a makeshift brush so that the pancake won't stick to the pan.
The agent of the action is unstated, but the construction of the sentence doesn't suggest it was the shallow pan that used the makeshift brush.
In order for the second part of the sentence to be a dangling modifier, it would have to be ascribing action to something that wouldn't normally make sense to be performing that action.
Using the same general sentence and reversing the order of its parts, we can reconstruct it in a way that does make use of a dangling modifier:
Using a makeshift brush so that the pancake won't stick to it, the shallow pan is first smeared with a tiny amount of edible oil.
What used the makeshift brush? Apparently, it was the shallow pan.
Although nobody would actually assume that to be the case, that's the objection that's made about the use of dangling modifiers.