Bradbury, who only turned professional in July, made two eagles in the space of three holes and also carded seven birdies at Houghton Golf Club to enjoy a one-shot lead over Germany's Nick Bachem.
The world number 1,397 completed his round shortly before play was suspended for the second time due to lightning and eventually called off for the day, with play due to resume at 6.45am local time on Friday.
"I got a little bit of momentum going on the back nine and the thunderstorm stopped that, but I can't really complain with a 63," the 23-year-old from Wakefield said.
"It's my first time playing at altitude and the ball is going miles. I had nine iron into both par fives for the two eagles but Keegan, my local caddie, has been great.
"It's nice and warm - I got sunburnt yesterday (Wednesday) and won't be doing that again - but it's just great to be out here and playing in some nice conditions."
Defending champion Thriston Lawrence carded a five-under-par 66 as he celebrated becoming the first South African player to be named rookie of the year on the DP World Tour.
Lawrence, who made his major championship debut in the 150th Open at St Andrews and also won the Omega European Masters, said: "It's a dream come true.
"If you look at the names on the trophy, it's incredible. A year ago I didn't even have a category, so when I started off with a victory it came to mind straight away to go for this award.
"To have accomplished it is an incredible feeling. I'm very grateful and honoured."