In what ways do air traffic controllers communicate with pilots in class A airspace?
Methods

VHF radio is one method. HF is used in oceanic areas, but it is in decline.

Data comms are also used, it's called CPDLC (Controller–pilot data link communications). The ATC sends a digital message to the pilot, and they respond the same way.

Too Much Static Noise (HF)

On long flights, the controller can grab the crew's attention by poking them using SELCAL. Too much static noise for a long time is tiring to listen to.

Chatter

Most of the chatter is handing off the plane from one controller to another.

If there's bad weather en-route, the pilot will ask to deviate. That's another type of chatter.

Also climb, step-climb, and descent instructions.

Changes to route are also relayed.

When flying over areas not covered by radar, position reports are also provided by the crew and confirmed by the controllers.

PIREP's—or pilot reports—often include information like "watch out there's icing, tell the folks behind me."

Traffic separation instructions and traffic advisory information are also quite common.