Prepare the Vegetables : Fill the large pot with about 6 cups of water and bring to a rolling boil. While it heats, trim the woody ends from the asparagus spears and cut the spears into 1 ½-inch pieces. Juice half of the lemon, reserving the other half for serving.
Blanch the Asparagus : Once the water is boiling, carefully add the asparagus pieces. Cook for 90 seconds until the color brightens and they are crisp-tender. Immediately use the slotted spoon to transfer the asparagus into a bowl of ice water for 1 minute to stop the cooking process. Drain thoroughly and set aside.
Whip Up the Arugula Pesto : In the bowl of your food processor, combine the arugula, walnuts, grated Parmesan, garlic cloves, the juice from half a lemon, and black pepper. Pulse several times to break down the ingredients. With the machine running, slowly stream in 1 tablespoon of the olive oil until a mostly smooth, vibrant pesto forms.
Achieve the Perfect Gnocchi Sear : Heat the remaining 1 tablespoon of olive oil in the skillet over medium-high heat until it shimmers. Arrange the frozen gnocchi in a single, even layer. Allow them to cook, undisturbed, for 5–7 minutes, until the bottoms are deeply golden-brown and release easily from the pan.
Crisp the Second Side : Flip the gnocchi using a thin spatula. Continue to cook for another 5–7 minutes until the second side is equally crisped and the gnocchi are cooked through.
Combine and Finish : Reduce the heat to low. Add the blanced asparagus and julienned sun-dried tomatoes to the skillet with the gnocchi. Stir gently for about 60 seconds to warm everything through. Remove the skillet from the heat, add the arugula pesto, and toss until every piece is evenly and generously coated. Serve immediately with a wedge of the remaining lemon.