Well, first thing I'd do is open up the "giant talking stalk of broccoli" to make sure the "supernatural" force inside isn't my brother-in-law.
If it's just a stalk of broccoli, I'd ask it why it's doing my lawn for free. (Better be a good answer too. The last thing we need are a bunch of communist broccoli stalks undermining free enterprise.)
To put the question more seriously, as long as something has an observable effect on the real world, what's to prevent it from being reachable through scientific investigation?
Nothing. But, unlike your example above, where you start with the assumption that the broccoli stalk is necessarily animated by "some supernatural force" before investigating the more mundane physical probabilities, science doesn't start its investigation with an assumption that there is no explanation other than the supernaturally inexplicable. If it did, not much would get explained, would it?
That wasn't the assumption I'd make. What I was saying in that hypothetical example is that a "supernatural" force actually was causing this to happen. That's not an assumption, it's a fact (something I have the luxury of imposing when coming up with hypothetical examples). So my question is that given that fact, would science be capable of recognizing it (regardless of what assumptions it starts out with initially)?