For the chicken: In a large bowl, combine the salt, brown sugar and 4 cups hot water; whisk until dissolved. Add the oregano and bay leaf and the ice, and stir until cold. Add the chicken breasts, cover, and refrigerate for at least 20 minutes and up to 45 minutes.
Heat a grill pan over medium-high heat. Remove the chicken breasts from the brine and pat dry; discard the brine. Grill the chicken breasts on the first side until they release easily from the grill pan and grill marks appear, about 4 minutes. Flip and cook for about 4 minutes on the reverse side. Flip again and cook 4 minutes. Flip a final time and cook 4 minutes. Remove the chicken to a cutting board and let rest for 10 minutes. Slice into thin strips.
For the slaw: In a medium bowl, combine the cabbage, jalapenos, carrots, scallions, snow peas and salt; toss to combine. Let sit for about 10 minutes so the salt begins to wilt the cabbage.
Meanwhile, in a small bowl, whisk together the olive oil, lime juice and cumin. Add the dressing to the salted vegetables; toss well. Add the cilantro and set aside.
For the beans: In a heavy-bottomed pot set over medium-high heat, cook the bacon until it begins to render its fat. Add the onions and sweat until translucent. Add the garlic and cook until fragrant. Add the black beans and the cumin, and lower the heat to medium-low. Cook for about 20 minutes, until the garlic begins to soften and the beans begin to break apart. Transfer to a food processor and puree until smooth. Season with salt to taste.
For the burrito: Combine the Greek yogurt, paprika and salt in a small bowl. Stir to combine.
Heat a cast-iron pan over medium-high heat. Warm the tortillas in the pan, one at a time, for about 30 seconds per side. Lay the tortillas flat on a work surface and build the filling in a tight pile in the center of each tortilla. To each, start with 3 ounces (about 1/4 cup) of the refried beans, followed by 2 ounces of the sliced chicken, about 1/2 cup of the slaw, one-quarter of the avocado slices, a dollop of the yogurt sauce and 1 tablespoon of the Monterey Jack.
To wrap the burritos: Hold a tortilla at the 5 o'clock and 7 o'clock positions and tightly fold in each side (5 and 7). Add 1 tablespoon of the remaining Monterey Jack, then fold the 12 o'clock side down over the cheese to form a triangle-shaped burrito. Flip the tortilla on its seam to keep the shape intact. Repeat.
Spray the hot cast-iron pan with cooking spray. Lay the burritos in the pan, seam-side down, and cook until sealed and nicely browned, about 3 minutes. Using a spatula, flip the burritos to brown on the reverse side. Remove from the pan once browned. Slice in half (forming two triangles). Serve immediately.