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Step 1
Pound your chicken breasts to 1/4 - 1/2 inch thickness between some plastic wrap or inside of a large ziplock baggie.
Step 2
Melt 1/2 stick of butter and 2 tablespoons of vegetable oil in a large skillet. Heat over medium heat until butter is melted. Add mushrooms and green onions. Brown the mushrooms. Add in the green onions during the last 2 minutes of cooking. Put everything in a small bowl and set aside.
Step 3
Add another 1/2 stick of butter to the pan. Add a little more more oil (if necessary). The oil helps keep the butter from burning. Also, start your large pot of boiling water for the pasta at this point.
Step 4
Combine flour, marjoram, thyme, salt, and pepper in a medium bowl. Lightly coat chicken breasts in flour and place in a single layer in frying pan over medium heat. Brown on both sides. Remove and repeat with remaining breast cutlets. Remove all when done, leaving behind the butter and brown bits in the pan. It is okay if there is a bit of pink in the middle. They will cook more in the sauce at the end.
Step 5
Add Marsala and let simmer for approximately 1 minute. Add chicken broth. Stir to combine and add in mushroom/onion mixture and chicken breasts. Stir a bit and then let simmer, covered, over medium-low heat. Cook until sauce is thickened, approximately 20 minutes.
Step 6
Brown the butter in a small pot by melting 1/2 stick of butter over medium heat. Don't touch it, but watch it carefully. It will start to bubble, foam, and under that foam it will start to brown. Brown well, removing from heat just before you think it'll burn.
Step 7
When pasta is done, drain well and toss with browned butter and grated parmigiano reggiano or mizithra cheese.
Step 8
Serve the chicken with some sauce poured over and a healthy heap of your brown butter pasta.