Cut the chicken breasts in half horizontally. Season liberally on both sides with salt and pepper
Heat oil in a large skillet on medium high until shimmering, add chicken breasts and fry until golden brown, about 3-5 minutes per side. You want to develop a good maillard, so press the breasts down
Remove the breasts from the pan, leaving the juices and set aside
Add more oil and the mushrooms, season with salt and sautee until darkened and softened- another 3-4 minutes
As the mushroom are approaching done add the shallots and thyme and continue to sautee another 1-2 minute.
Add the white wine and scrape the bottom, allow to come to a simmer, then add butter
Once butter is melted, whisk in the flour until fully combined
Add the broth and cream and bring to a simmer, reduce heat to low, return the chicken to the pan and simmer uncovered for 10-15 minutes or until sauce is desired consistency