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The 15-year-old clocked one minute 50.23 seconds over her two runs, 0.96 seconds ahead of Germany's Hannah Neise.
Pittaway's triumph follows fellow Brit Madi Rowlands, who won gold in the ski halfpipe on Sunday in Lillehammer.
Rowlands, 15, added a bronze in the women's ski slopestyle final on Friday.
"When I first crossed the line I couldn't believe it," said Pittaway.
"I felt like I was standing next to me watching me finish - it was so surreal and just like a movie.
"Hannah [Neise] is one of my best friends and we're really happy to be on the podium together. That's been our dream together for four years."
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The wind farm, run by community company Viking Energy, would be one of the biggest in Scotland.
Sustainable Shetland, a group opposed to the development, launched a judicial review.
The challenge, at the Court of Session in Edinburgh, has been adjourned. It is expected to resume at the end of April.
Energy Minister Fergus Ewing granted consent for the scheme in April.
Protesters claim the development is too big and would blight the landscape.
Supporters argue it would raise money for the islands, create jobs and help meet renewable energy targets.
The 370MW wind farm is aimed at powering more than 175,000 homes despite Shetland having a population of about 22,000.
It is estimated the wind farm could bring about Â£30m annual income for the local community.

He accepted that he made contact with Sophie Cahill's face when he flailed his arms during a scuffle.
But he told Croydon Magistrates' Court it was accidental.
Ms Cahill, of Cwmbran, Torfaen, said Mr Bowers had hit her in the face, bloodying her nose, and had pulled her hair and squeezed her throat.
Asked by John Blandford, defending, if it was true that he had beaten her up, the former Another Level singer said: "No, of course not."
The court has been told that the incident in January happened after Ms Cahill quizzed him about why he had glitter on his face.
She became suspicious after spotting it as she tried to wake Mr Bowers. The couple had already split up but were living separately in the home they shared with Bowers' parents in Croydon, south London.
Ms Cahill said after she pointed out the glitter, he launched an attack in which he grabbed her by the shoulders, threw her out of a bedroom and hit her in the face, giving her a bloody nose, pulled her hair and squeezed her throat.
Mr Bowers, 35, who denies one count of assault by beating, said he had been to a circus-themed nightclub the previous evening because he was trying to help a friend get a job.
He says he got home at 02:00 or 03:00 GMT and spent time talking with his father.
The trial is ongoing.
Ashleigh Pittaway claimed Great Britain's second gold medal of the Winter Youth Olympics, winning the skeleton bob event in Norway.
[NEXT_CONCEPT]
A legal challenge to plans to build a 103-turbine wind farm in the centre of Shetland has been adjourned.
[NEXT_CONCEPT]
Pop singer Dane Bowers has denied assaulting his ex-girlfriend, saying he was acting in self-defence when he hit her.