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.: The first smell you will notice is the clean, steamy aroma of boiling water, and the sound is a steady gentle roar from the pot; this tells you the heat is high and consistent. Cooking the pasta until it is al dente gives a slight resistance when you bite, which keeps the texture pleasant once it mixes with the other ingredients. Reserve 1 cup of the starchy pasta water because it will become a silky binder when tossed with the goat cheese , creating a creamy coating without heavy cream. After draining, toss the pasta with a little olive oil so it does not clump, and set it aside on a warm surface to maintain heat. A common mistake is overcooking the pasta which leads to mushy noodles once combined; test a strand early and often toward the end of the package time to avoid that.
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.: You will hear a lively sizzle as the chicken hits the hot olive oil , and the smell of browned meat will develop quickly, signaling flavor building through the Maillard reaction. Searing at medium-high creates a golden surface that locks in juices, giving tender slices that hold up in the final toss. When the chicken reads 165 degrees F on a thermometer or is opaque throughout, remove it with a slotted spoon to keep any cooking juices in the pan. Place the cooked chicken on paper towels to absorb excess oil. Be careful about oil splatter when adding the chicken ; it can pop if the pan is too hot or the meat is wet, so pat the pieces dry first. Overcrowding the pan is another pitfall, which prevents proper browning, so work in batches if needed.
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).: The fragrant sweetness of the onion will become apparent almost immediately as its edges become translucent, releasing a gentle savory scent that complements the browned chicken . Adding the peas next introduces little pops of sweetness and bright green color; frozen peas will steam and soften slightly longer than fresh. This step builds layers of flavor in the pan and gives the dish vegetal brightness. A common mistake here is leaving the heat too high so the onion chars rather than softens; lower to medium to coax sweetness without burning.
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.: At this point the pan will present a lovely mixture of aromas, with citrus from the lemon zest cutting through the savory base. Adding the reserved starchy water, if needed, helps the components bind into a glossy sauce because the starch emulsifies with the added olive oil and goat cheese later. The visual cue to watch for is a cohesive, slightly glossy coating on the pasta and pieces, not a puddle of liquid. Overdoing the water can dilute flavor, so add it conservatively to reach a saucy but not soupy texture.
Just before serving, stir in the spinach, dill, and goat cheese. Serve warm.: When you add the handfuls of fresh spinach , it will wilt quickly into the warm pan, releasing a mild green scent and soft texture that blends into the pasta . Crumbling in the goat cheese creates pockets of creamy tang that melt into the strands when stirred, forming a silky coating that carries the herb and citrus flavors. Finish with the chopped dill for a fresh, aromatic top note. Serve while warm so the goat cheese remains luscious. A common error is adding the spinach too early which causes it to overcook and lose brightness, so fold it in at the end for the best texture and color.