[1] Slama began his professional career in 2007 with the rookie-level Elizabethton Twins, and after only six appearances where he struck out 10 in 7+1⁄3 innings, earned a promotion to the Single-A Beloit Snappers.

He compiled a 2-0 record with a 0.50 earned run average and 12 saves in the first half of the 2008 season—helping Fort Myers capture the Florida State League first-half West Division title.

Along with teammates Robert Delaney, Brian Dinkelman, Jeff Manship, Wilson Ramos, and Danny Valencia, Slama represented the Miracle in the 2008 Florida State League All-Star game.

For the season, he went 4-1 with a 1.01 ERA, 25 saves and 110 strikeouts in 71 innings to earn All-FSL Team honors, the Jim Rantz Award Twins' Minor League Pitcher of the Year and MILB.com's "Best Class A Reliever.

"[citation needed] He recorded the save in both division play off games against the Dunedin Blue Jays, and in the Miracle's one win against the Daytona Cubs in the FSL championship series.