Season chicken with salt and pepper.
Heat olive oil in a large skillet over medium-high heat. Add chicken and cook for 5 minutes.
Reduce heat to medium and add onion; cook until translucent and soft, about 3 minutes. Add mushrooms and cook, stirring occasionally, for 4 to 5 minutes. Stir in garlic and sauté until fragrant, about 30 seconds.
Pour in chicken broth, wine, and sherry and bring to a boil. Stir in Greek yogurt, 1/4 cup dill, and Dijon mustard until well incorporated. Add butter and stir until melted.
Place flour in a small bowl. Add a ladle of hot broth and whisk until smooth and no lumps remain. Pour mixture into the skillet and stir until well incorporated.
Reduce heat to low and simmer until the sauce starts to thicken, 3 to 4 minutes. Taste and adjust seasoning and/or add additional dill if necessary.
I use Fage® Total 2% yogurt.
If you don't have fresh dill, you can use dried dill, just don't use as much, since it's really concentrated. A fresh to dry ratio is usually 3:1.