Preheat oven to 450°F. In a skillet over medium heat add the oil until it shimmers and gets warm. Add in the mushrooms and shallots. Cook for 5-6 minutes or until the mushrooms have released their liquids and start to lightly brown. Add in the garlic, stir, and cook for 1 minute. Add in the arugula (may have to add in batches) and gently stir to wilt. Once it’s all wilted add in 1 teaspoon salt, 1 teaspoon pepper, and 1/2 teaspoon thyme. Remove from the heat and mix. Transfer the mixture to a strainer (save the liquids for a pan sauce) and allow to drain.
Halve chicken breasts horizontally, keeping 1 long side attached; open halves like a book (butterfly). Pound breasts until 1/4″ thick. Equally divide the mixture along the lower third of the breast, maintaining a 1/2″ border. Roll up the chicken lengthwise and tie it with kitchen twine. Repeat with remaining chicken breasts.
Heat 1 tablespoon of butter and oil in a large heavy ovenproof skillet. Add roulades and cook until browned on all sides, 8-10 minutes. Transfer skillet to oven and bake until a thermometer inserted into the center of roulades registers 165°F, 7-8 minutes. Transfer chicken to plates, lightly tented with foil, and let rest for 10 minutes.
Set skillet over medium-high heat and melt remaining butter. Scrape up the bits at the bottom of the pan with a wooden spoon. Add in the chicken stock, salt, pepper, and thyme, and bring to a boil. Simmer until liquid is thickened and reduced, 10-12 minutes. Whisk in the cream for a more cream-based sauce. Adjust seasonings. Cut the twine off of the chicken, slice into 1/2” slices, and drizzle with sauce.