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Heat your oven to 275 degrees.
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First you want to bread and sauté the chicken in batches. Melt about half of a tablespoon of butter and a splash of olive oil in a skillet over medium heat. Once the pan is hot, start breading the chicken and adding it to the pan. Take the chicken breasts and coat them in the breadcrumbs, then the eggs, and then coat them in a second layer of breadcrumbs. Brown each side of the chicken, adding more butter to the pan as necessary. Continue browning the chicken in batches, adding more butter and olive oil to the pan as you go.
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Once the chicken has browned, put it in the oven at 275 degrees for 30 minutes to finish cooking.
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Once the chicken has been baking for 15 minutes, prepare the sauce and the pasta. Put a pot of water on for the pasta, while you are waiting for the water to boil, get to work on the sauce.
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To make the sauce, melt 1 tablespoon butter in a skillet over medium heat. Once the butter has melted, add the mushrooms. Season with salt and pepper and stir to evenly coat the mushrooms. The mushrooms will still be quite dry, but that is ok. Let them cook down in the pan for 5 minutes, stirring frequently. Then add the white wine and let simmer for 3 minutes. Then add another tablespoon of butter, plus the chicken broth, lemon juice, and capers. Bring to a steady simmer and let reduce for 5 minutes.
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While the sauce is simmering, cook the pasta.
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Serve the chicken on a bed of pasta with the mushrooms and sauce spooned over.