Wash and dry all produce. Bring a large pot of salted water to a boil. Place peppercorns in a zip-close bag; crush using a heavy-bottomed pan or a rolling pin. Zest 2 tsp zest from lemons and then halve; squeeze 3 TBSP juice into a small bowl. Mince garlic.

Once water boils, add linguine to pot. Cook until al dente, 9-11 minutes. During the last minute of cooking, stir in spinach and let wilt. Reserve 1 cup cooking water, then drain linguine and spinach. Set aside.

Meanwhile, pat chicken dry with a paper towel; season all over with salt and a pinch of crushed peppercorns. Heat a large drizzle of olive oil in a large pan over medium-high heat. Add chicken and cook until browned on surface and no longer pink in center, 6-8 minutes per side. Transfer to a cutting board and let rest 5 minutes.

Heat a drizzle of olive oil in pan used for chicken over medium-high heat. Add garlic, 11/2 tsp lemon zest, and 1 tsp crushed peppercorns. Cook, stirring, until fragrant, about 30 seconds. Stir in reserved lemon juice and half the reserved cooking water. Bring to a simmer, then remove from heat.

Thinly slice chicken once it has had 5 minutes to rest. Using tongs, toss linguine, spinach, and chicken into pan. Add 4 TBSP butter and half the Parmesan, stirring to melt. Season with salt. TIP: Everything should be coated in a loose sauce—if dry, add more cooking water a little bit at a time.

Divide pasta mixture between plates. Garnish with remaining Parmesan, remaining lemon zest, and any remaining crushed peppercorns (if desired—you may want to skip this for the kids).