Season chicken with 1/2 teaspoon salt and pepper. Heat oil in a large skillet over medium-high heat and brown chicken for 2 minutes per side, then transfer to a 5-quart slow cooker.
Cook mushrooms and shallots in the skillet for 8 to 10 minutes until mushrooms soften and their liquid mostly evaporates. Add garlic for 30 seconds.
Pour in Marsala and simmer for 3 minutes, scraping the bottom. Add broth and remaining salt, then pour everything over the chicken.
Cover and cook on low for 2 1/2 to 3 1/2 hours, checking from 2 1/2 hours and removing breasts as they reach 165°F. Transfer chicken and mushrooms to a serving dish and cover.
Pour cooking liquid into a saucepan and bring to a simmer. Mix cornstarch with cold water, whisk into the liquid and cook for 2 to 3 minutes until lightly thickened.
Switch off the heat and stir in butter. Spoon the sauce over the chicken and mushrooms, serving one breast per person.