Bring a large pot of salted water to a boil and cook pasta until al dente, according to package directions. Reserve 1 cup of the pasta water, then drain. Toss the drained pasta with a bit of olive oil to prevent sticking, then set aside.
Heat 1 tablespoon olive oil in a large skillet over medium-high. Once hot, add the chicken and sauté until cooked through, 4 to 6 minutes (be careful as the oil may splatter). With a slotted spoon, remove to plate lined with paper towels and set aside.
Reduce pan heat to medium. Add 1 teaspoon olive oil to the skillet, the onions, salt, and pepper, and cook until fragrant and beginning to soften; about 1 minute. Add the peas and cook until heated through (about 1 minute for fresh or 3 minutes for frozen).
Add the reserved chicken, reserved pasta, remaining 2 teaspoons olive oil, lemon zest, and lemon juice. If the pasta seems to dry, add a little of the reserved pasta water.
Just before serving, stir in the spinach, dill, and goat cheese. Serve warm.
TO STORE: Place leftover pasta in an airtight storage container in the refrigerator for up to 3 days.
TO REHEAT: Gently rewarm pasta in a large skillet on the stove over medium-low heat, adding splashes of broth as needed to thin the sauce. Top leftovers with a squeeze of lemon juice to refresh the flavors. You can also reheat this dish in the microwave.