In a large skillet over medium-high heat, combine the butter and olive oil. Season with salt and pepper to taste. Place the chicken and cook until they are no longer pink inside. Time will vary, depending on the size of the chicken tenders.
Remove the chicken tenders from the skillet. Add the mushrooms and onions cook for about 8 to 10 minutes or until slightly browned and tender. Season with salt and pepper to taste.
Remove the mushrooms and onions from the skillet.
Add the butter and flour to the skillet. Stirring constantly until the butter is melted and the butter and flour form a paste.
Slowly add the chicken broth, whisking so lumps do not form. Once the broth is added continue to cook until the mixture starts to thicken.
Taste to see if you need to season with salt and pepper to taste.
Add the chicken tenders and mushrooms back to the skillet and simmer for about 10 minutes.
Serve.