Cook pasta according to directions in heavily salted water. Three minutes before it's al dente, add the frozen cut green beans. Cook for another two minutes, until green beans are tender and just before pasta is al dente.
Meanwhile, mix the flour (2 tablespoons) with the kosher salt (1/2 teaspoon) and black pepper (1/4 teaspoon) in a large bowl. Toss the cut up chicken breasts (1 lb.) in the flour mixture to completely coat.
Heat the 2 tablespoons of olive oil and 2 tablespoons of the butter in a large skillet. Add the flour-coated chicken pieces to the skillet and cook for about 5 minutes, stirring occasionally, until browned and fully cooked. Remove from skillet onto a plate and set aside.
Add the onions to the skillet and sauté until softened and starting to brown (about 2 minutes). Add the garlic and sauté until fragrant; about 30 seconds. Add the white wine (1/2 cup) and use a a wooden spoon to scrape up any stuck-on bits from the bottom of the skillet.
Add the chicken broth (1 cup), lemon juice and zest, and capers (2 tablespoons), and simmer until liquid has reduced by about half.
Use a slotted spoon to transfer the pasta and green beans to the skillet, along with 1/2 cup of the pasta water, and stir to coat. Cook until pasta is fully al dente and sauce has thickened, about 2-3 minutes. Turn the heat to low and stir in the remaining 2 tablespoons of butter until melted. Add the chicken back to the skillet and stir until heated through.
Serve with extra lemon slices, parsley, and grated parmesan, if desired.
The sauce will seem thin at first. Don't worry! Keep cooking it to thicken for a few minutes at the end, and if it still seems thin, allow it to cool for a while before serving, which should help solidify it a bit. Adding grated parmesan cheese is another way to thicken similar sauces- the cheese melts into it as it cooks.
The provided nutrition information does not include any added sodium from seasoning to taste, any optional ingredients, and it does not take brands into account. Feel free to calculate it yourself using this calculator or by adding the recipe to Yummly.