Using an immersion circulator according to product directions, heat a water bath to 145°F. Meanwhile, in a 12-inch skillet over medium-high, heat the oil until shimmering
Add the chilies and toast until lightly browned, about 20 seconds per side. Transfer to a food processor, reserving the oil in the skillet. Process until coarsely chopped, about 30 seconds
In a small saucepan, bring the water to a boil. Add the chilies, oregano and garlic then cover, remove from heat and let sit for 15 minutes
In the food processor, combine the sugar, vinegar, 2½ teaspoons salt, ¼ teaspoon pepper, cumin, cinnamon and 2 tablespoons of the reserved chili oil
Add the chili-water mixture and process until smooth, about 1 minute, scraping the bowl as needed
Place each chicken breast in a quart-size zip-close plastic bag. Add 1/4 of the chili sauce to each bag, then seal the bag and shake to coat evenly
Vacuum seal the bags, then set into the water bath. Once the water returns to 145°F, let the chicken cook for 1½ hours
When the chicken is done, remove from the bags to a serving platter. Transfer any liquid in the bags to a 12-inch skillet and bring to a simmer over medium
Cook until thickened and reduced, about 1 minute. Whisk in the butter and lime juice. Serve the chicken drizzled with the reduced sauce and sprinkled with cilantro