Combine oil, garlic, thyme sprigs, and tarragon sprigs in a small saucepan over medium heat. Cook, stirring occasionally, until garlic just begins to brown, about 4 minutes. Remove from heat; let stand 10 minutes. Discard herb sprigs.
Using a spiralizer, cut zucchini and carrots into long “noodles.” Combine zucchini, carrots, pepper, and 1/2 teaspoon of the salt in a medium bowl; toss to coat. Let stand 10 minutes; drain.
Cook pasta according to package directions, omitting salt and fat. During last 6 minutes of cooking, place whole eggs in boiling water with pasta. Drain pasta and eggs. Plunge eggs into a bowl of ice water; let stand 5 minutes. Add pasta, oil mixture, and remaining 1/2 teaspoon salt to zucchini mixture; toss to coat.
Divide pasta mixture evenly among 4 plates. Peel eggs and cut in half lengthwise. Top each serving with 2 egg halves. Sprinkle servings evenly with cheese, chopped tarragon leaves, and chopped thyme leaves.