Place chicken breasts in the slow cooker insert. Pour in chicken broth and sprinkle with cumin, garlic powder, 1/2 tsp salt, and 1/4 tsp pepper. Cover and cook on low for 3-4 hours or on high for 1.5-2 hours, until chicken is easily shredded with two forks. Remove chicken, shred, and set aside. Reserve about 1/4 cup of the cooking liquid from the slow cooker.

While the chicken cooks, preheat your oven broiler to high. Place poblano peppers on a baking sheet. Broil for 5-7 minutes per side, turning with tongs, until the skins are blistered and charred. Alternatively, you can char them over an open flame on a gas stove burner. Transfer the roasted poblanos to a bowl and cover tightly with plastic wrap or place them in a paper bag for 10-15 minutes to steam. Once cooled, peel off the skins, remove the stems and seeds, then slice the peppers into strips or large pieces. Set aside.

Heat olive oil in a large skillet over medium heat. Add the diced yellow onion and cook until softened, about 5-7 minutes. Add the minced garlic and cook for another minute until fragrant, being careful not to burn the garlic. Remove from heat and set aside.

In a large mixing bowl, whisk together the 4 large eggs, heavy cream, 1/2 tsp salt, 1/4 tsp black pepper, and dried oregano. Stir in the undrained diced tomatoes with green chilies, undrained diced mild green chilies, and the reserved 1/4 cup chicken cooking liquid from the slow cooker. Mix well to combine.

Lightly grease the slow cooker insert. Spread half of the roasted poblano strips evenly on the bottom of the slow cooker. Top this layer with all of the shredded chicken, then sprinkle with half of the Monterey Jack and cheddar cheeses. Evenly spread the sautéed onion and garlic mixture over the cheese.

Pour half of the egg mixture evenly over the layers. Arrange the remaining poblano strips on top, then sprinkle with the rest of the shredded cheeses. Finally, pour the remaining egg mixture evenly over the top, ensuring it covers all ingredients as much as possible.

Cover the slow cooker and cook on low for 3-4 hours, or on high for 1.5-2 hours, until the egg mixture is set, the casserole is bubbly, and heated through. The edges should appear golden brown and the center should be firm to the touch.

Once cooked, turn off the slow cooker and let the casserole rest for 15-20 minutes before serving. This allows it to set further, making it easier to slice and serve. Garnish with fresh chopped cilantro, if desired, and serve warm.