Slice the beets into 1/8” slices, then cut into julienned strips. Transfer to a bowl and set aside.
Heat a large, heavy skillet over high heat. Add the oil, caraway seeds, and sliced beets. Sauté 10-12 minutes, stirring occasionally, until the beets look dark around the edges and lose their moisture. Remove from heat and set aside.
Bring 6 quarts of water to a boil in a large pot. Add tablespoon of salt, then add the pasta and cook according to package directions, until al dente. Drain, reserving ½ cup of the pasta water.
Heat the beets over high heat. Stir in garlic and vinegar, and sauté for 1 minute. Add the drained pasta, reserved cooking water, and tarragon. Sprinkle with remaining ¾ teaspoon of salt and a pinch of black pepper. Remove from heat and use tongs to toss the pasta until well-coated with sauce. Serve immediately.