How to Make It
Cook chicken, poblano chiles, onion, and garlic in a Dutch oven over medium-high until meat crumbles and is no longer pink, 10 to 12 minutes. Stir in tomato paste, chili powder, and cumin, and cook, stirring constantly, 1 minute. Stir in black beans, tomatoes, and chicken stock; bring to a boil. Reduce heat to medium-low, and simmer, stirring occasionally, 20 minutes.
Stir in squash, and cook until squash is just beginning to soften, about 5 minutes. Stir in 1 teaspoon of the salt and 1/2 teaspoon of the black pepper. Transfer chili to 6 lightly greased 10 oz. ramekins (or a 3-quart baking dish).
Preheat oven to 400°F. Bring water and 1 teaspoon of the salt to a boil in a large saucepan over high; whisk in grits, and cook, stirring often, until tender and thickened, about 25 minutes. Remove from heat, and whisk in cheese and cilantro. Stir in remaining 1 teaspoon salt and 1/2 teaspoon black pepper. Gradually stir about one-fourth of grits mixture into eggs; add egg mixture to remaining hot grits mixture, stirring constantly. Spoon hot grits mixture over chili.
Bake in preheated oven until crust is set and golden, 22 to 25 minutes for ramekins or 30 to 35 minutes for baking dish. Let stand 10 minutes. Garnish with cilantro leaves, if desired.