Using a vegetable spiralizer, turn zucchini lengthwise into long “zoodles”. (When you reach the seeds in the middle, stop spiralizing as seeds make the noodles fall apart.)
Place the zoodles into a colander and toss with a bit of kosher salt. Let zoodles drain for 15 to 30 minutes, then gently squeeze to remove any excess liquid.
Meanwhile, heat butter and 1 Tbs. oil in a large skillet over medium-high heat. Add garlic and cook, stirring, for 30 seconds.
Whisk broth and cornstarch in a small bowl; add to garlic in skillet, along with shrimp, wine, and lemon juice. Simmer, stirring occasionally, until shrimp are just cooked through, 2-3 minutes. (Shrimp will turn from gray in color to pink very quickly when done!) Immediately remove skillet from heat.
In a separate skillet, heat the remaining 1 Tbs. olive oil over medium-high heat; stir in pesto, then add the zoodles and gently toss, cooking just until al dente, 2-3 minutes.
Serve the prepared shrimp with sauce over the cooked zoodles at once. Enjoy!