Pat fully thawed breasts dry, halve an unusually thick piece horizontally and season with 3/4 teaspoon salt and pepper.
Add oil, shallot, garlic, 1 cup broth, thyme, oregano and rosemary to a 4- to 6-quart slow cooker.
Place chicken in one layer with minimal overlap, spoon broth over exposed surfaces and cover.
Cook on high 2 to 3 hours, beginning checks at 1 hour 45 minutes for thin pieces.
Remove each breast when its thickest center reaches 165°F. Rest on a clean plate for 5 minutes.
Boil cooking liquid in a saucepan for at least 1 minute. Whisk cornstarch with remaining cool broth and add gradually until sauce lightly coats a spoon.
Remove from heat. Add lemon juice and adjust salt; whisk in cold butter, then zest and parsley without returning to a boil.
Slice rested chicken across the grain and spoon warm lemon garlic sauce over it.
