Bring a large pot of salted water to a rolling boil over high heat.
Add spaghetti to boiling water and cook stirring occasionally until al dente, about 9 minutes.⏱ 9 min
Whisk whole eggs, egg yolks, grated Pecorino Romano, and black pepper in a mixing bowl until smooth.
Cook diced guanciale in a 12-inch skillet over medium heat until fat renders and pork is crisp, about 7 minutes.⏱ 7 min
Ladle out half a cup of starchy pasta water from boiling pot before draining noodles.
Use tongs to transfer hot pasta directly into skillet with crisped pork and rendered fat.
Remove skillet from heat completely and let cool for 1 minute so eggs do not scramble.⏱ 1 min
Pour egg and cheese mixture over warm pasta while tossing constantly with tongs.
Splash in reserved warm pasta water a little at a time while tossing until a glossy sauce coats noodles.
Divide pasta immediately into shallow bowls and garnish with extra grated cheese and cracked pepper.
Preheat oven to 425 degrees Fahrenheit and place a sheet pan inside to warm up.
Pat chicken thighs completely dry with paper towels so skin crisps properly in the oven.
Toss potato chunks with half the olive oil, half the garlic, salt, and pepper in a bowl.
Rub remaining olive oil, garlic, oregano, lemon zest, salt, and pepper evenly over chicken thighs.
Carefully spread seasoned potatoes across the hot preheated sheet pan in one single layer.
Place chicken thighs skin-side up among potatoes and lay lemon rounds over vegetables.
Roast at 425 degrees Fahrenheit until chicken skin is golden and potatoes are tender, about 35 minutes.⏱ 35 min
Check chicken internal temperature reaches 165 degrees Fahrenheit using an instant-read meat thermometer.
Transfer to plates and let chicken rest 5 minutes before serving to keep meat juicy.⏱ 5 min