Twice, in 1991 and 1998.
Tom Glavine has won two Cy Young awards in his career. He won the award in 1991 and 1998 while playing for the Atlanta Braves.
He never won the Award
He won the Award 3 times, 1973, 1975 and 1976.
He won the Award 3 times, 1963, 1965 and 1966.
He won the Award 3 times, 1969, 1973 and 1975.
He won the Award 4 times, 1972, 1977, 1980 and 1982
4 times from 1992 to 1995.
Once in 1962
Once in 1971
Once in 1957
Once in 1959
Twice