Just another option, for the sake of showing different options:
(?<=^//)[^/]++
The server name will be in \0
or $0
or simply the result of the function, depending on how you call it and what your language offers.
Explanation in regex comment mode:
(?x) # flag to enable regex comments
(?<= # begin positive lookbehind
^ # start of line
// # literal forwardslashes (may need escaping as \/\/ in some languages)
) # end positive lookbehind
[^/]++ # match any non-/ and keep matching possessively until a / or end of string found.
# not sure .NET supports the possessive quantifier (++) - a greedy (+) is good enough here.