Using small sharp knife, cut stem from top of each fig, then cut straight down from top center to make 3/4-inch-deep, 1/2-inch-long slit in fig. Fill slit in each fig with cube of feta cheese, then press opening closed.
Place 1 slice of pastourma on work surface; top with sage leaf.
Place stuffed fig at 1 end and roll up to enclose fig. Secure with toothpick. Repeat with remaining pastourma, sage, and figs. DO AHEAD: Can be made 1 day ahead.
Arrange on plate, cover tightly with plastic wrap, and chill.
Let stand at room temperature 1 hour before continuing.
Pour enough oil into large skillet to cover bottom.
Heat over medium-high heat.
Add figs. Cook until pastourma begins to crisp, about 2 minutes per side.