Start by boiling water in a large pot. Toss in the thin spaghetti and cook until tender, about 8-10 minutes. Taste to check for doneness, ensuring it does not get mushy.
In a 12-inch skillet, heat olive oil over medium heat. Season the chicken cutlets with salt and pepper, then add them to the skillet. Sizzle until golden brown, about 4-5 minutes on each side. Cooking just until done is key, to avoid dryness.
Once the chicken is cooked, add finely minced shallots and garlic to the skillet. Cook until aromatic, about 1-2 minutes, watching carefully to prevent burning.
Carefully add the white wine to the pan, allowing it to simmer and reduce for 2-3 minutes to meld the flavors together.
Stir in the chicken broth and fresh lemon juice, letting the mixture simmer gently. The sauce should thicken slightly, about 3-5 minutes, while avoiding boiling too hard.
Toss in the drained capers and sliced lemon, stirring to incorporate. Taste and adjust seasoning, noting that capers can add saltiness.
Stir in the unsalted butter one tablespoon at a time until melted and combined into the sauce, making it glossy and rich. Do not let it sit too long over the heat.
Drain the spaghetti and add it directly to the skillet with the sauce. Gently toss everything together until the pasta is well coated, adding a splash of reserved pasta water if it seems dry.
Plate the pasta and garnish with freshly chopped parsley, ensuring the dish has a bright visual appeal.