Combine cider, salt and 1/2 cup sage, stirring until salt dissolves. Place pork in plastic zipping bag; add cider mixture. Seal bag; chill 12 to 24 hours.
Remove pork from brine and pat dry with paper towels. Sprinkle pork with pepper. Heat oil in a large skillet over medium-high heat; add pork.
To make Sage and Pecan Pesto:
Combine all ingredients in a food processor; process until ingredients are finely chopped. Makes 3/4 cup.