Cook the spaghetti according to the package directions, reserving ¾ cup of the cooking water. Drain the pasta and return it to the pot.
Meanwhile, heat the oil in a large skillet over medium-high heat. Add the shrimp, radishes, ½ teaspoon salt, and ¼ teaspoon pepper. Cook, tossing occasionally, until the shrimp are opaque, 4 to 6 minutes.
Add the shrimp and radishes, butter, tarragon, lemon zest, and ½ cup of the reserved cooking water to the pasta and toss to coat. (Add more cooking water as needed to loosen the sauce.)