> One could sell June $385 call options, currently trading at $17.

Assume one did this today. The closing price was $385.40. Will this short option get exercised Monday morning?

Technically possible, but almost certainly not.
The holder of that option could probably sell it for $18, so why would they exercise it to make only a 40 cent profit?
Options are almost never exercised for so long as there is a bid price which exceeds what the holder would make from exercising it.

Is it smarter to roll this contract to 390 if possible first thing Monday instead?

No reason you have to. I did exactly this, and I won't do so. I'm fine with it being in a mark-to-market loss for a while. As mentioned, nothing is likely to happen any time soon. Each day that goes by you're making a little money on time value, plus or minus squiggles depending on the stock price movement. And of course even if it were exercised, I'd just take my profit and buy the stock back again next time I like the asking price.

What I have done sometimes is this:
I think the stock is just starting to get a bit rich, so I write a call only a few bucks above the current price, say 3-6 months out. A bit of time goes by, but the stock soars some more, well above the strike. My positions are in a loss position mark-to-market basis, probably quite a bit on a percentage basis. (always remembering that the stock backing the call is also rising in price...it isn't a loss per se. Depending on how high the stock goes, it's a small net profit having done the call or a small profit combined with a missed opportunity to have sold even higher). But...now I think maybe the rally will continued. The time value of my call will have fallen, partly because of the time elapsed so far and partly because time value falls quickly as the stock price moves away from the strike. So, sometimes I close that one for a loss and write a new one, 3 months further out, for a premium which is usually somewhat close to the cost of closing the first position. This is definitely at a much higher breakeven if the stock is ultimately called away, and may also get me a bit more net time value I can profit from.
This habit is the main reason I don't usually start with long-dated calls when I'm writing them. Frequently I don't just write one, it ends up being a string of them (because I didn't call the top!), until the stock price peaks and pulls back again and I make most of the aggregate profit on the last one.

Jim