Is it possible to see the same sunset twice from an aircraft? If I initiate a climb just after watching the sun go down below the horizon, how fast would I have to climb in order to watch the sun come back up, so that I could then level off and watch the sunset for a second time?

I've Googled the answer, and several different articles have said that I'd need a minimum rate of climb of 5000 ft/min. However, I've also read several articles that said you can do the same thing in the Burj Khalifa's elevator, and it maxes out at 2000 ft/min. So, which of these is correct, if any?
Concorde did something similar: it would take off from London at sunset and then fly West faster than the Earth's rotation, so you would get a sunrise in the West on board.

This capability was also used for solar eclipse flights. While a solar eclipse lasts no more than 8 minutes on the ground, Concorde was able to stay in the Moon's shadow for 74 minutes.