Heat 2 teaspoons of the oil in a non-stick skillet over medium heat until shimmering. Add the leeks and cook until softened and beginning to brown at the edges, about 5 to 7 minutes. Stir in the garlic and thyme and cook until fragrant, about 30 seconds.

Slowly whisk in the broth and bring to a simmer, scraping up any browned bits. Remove from heat; stir in the peas and lemon juice. Let stand until the peas are heated through, at least 2 minutes. Season with salt and pepper to taste.

Meanwhile, bring a large pot of water to a boil. Add the asparagus and 1 teaspoon salt and cook, stirring often, until the asparagus is crisp-tender, 2 to 4 minutes. Remove the asparagus with a slotted spoon and set aside.

Return the water to a boil, add the pasta, and cook according to the package directions until al dente. When the pasta is finished cooking, reserve ¼ cup of the pasta water, then drain the pasta and return it to the pot.

Add the arugula to the pasta and toss to wilt. Add the leek-broth mixture, asparagus, goat cheese, and remaining 2 teaspoons of oil, toss to combine. Before serving, add the reserved cooking water as needed to loosen the sauce.