Should I call a plumber or a restoration company first?

If water is still flowing and you cannot shut it off at a valve: plumber first — the leak must stop. If the water is off (or the source is stopped): restoration first — every hour of saturation grows the loss, and the pipe repair can happen in parallel. They are different trades: the plumber fixes what failed; restoration fixes what the water did to your house.

In practice you often need both the same day, and they do not compete — we work alongside plumbers constantly (many of our jobs arrive as plumber referrals). The plumber replaces the burst section in an hour; the drying of the wall cavities, subfloor, and flooring that got soaked is the multi-day discipline with meters and logs.

One caution: a plumber saying "it will dry on its own" is talking about the pipe area he can see. Whether your wall cavities and subfloor dry on their own is a moisture-meter question, and it is the difference between a small loss and a mold problem next month.