Cook pasta in a medium pot of boiling salted water, stirring occasionally, until al dente, then drain.
Meanwhile, remove ribs and stems from chard; slice very thinly. Set aside.
Coarsely chop leaves; set aside.
Heat 1/3 c olive oil in a large skillet over medium-high. Cook chard ribs and stems, stirring occasionally, until lightly crisped, about 4 minutes. Add garlic, pistachios and remaining 1/3 c olive oil and cook, stirring often, until nuts are golden brown and garlic is softened, about 3 minutes.
Add sage; cook, stirring, until crisp, about 30 seconds. Remove pan from heat, add reserved chard leaves, and stir until wilted and combined.
To the chard mixture, stir in vinegar, capers, sun-dried tomatoes, red pepper flakes, salt and pepper. Add pasta, feta and toss to combine before serving.