Preheat the oven to 350 degrees F.
For the grillades: Cut the steak in half. Sandwich each half between 2 pieces of plastic wrap or wax paper, and pound to an even 1/8-inch thickness. Sprinkle the steak on both sides with salt and pepper, and cut into 2-inch pieces.
Mix the paprika, New Mexican chile powder, garlic powder, onion powder, oregano and chile de arbol powder in a small bowl until combined. Put the flour in a shallow baking dish and season it with salt and pepper.
Heat the butter and canola oil in a large, high-sided saute pan over high heat. Working in batches, coat the steak pieces in the seasoned flour, shaking off the excess, and brown them on both sides in the hot fat. Transfer the steak to a plate as you finish each batch. Add the onions, poblano and red pepper to the pan, adding more oil if needed, and cook until the vegetables are very soft, about 5 minutes. Add the garlic and cook for 1 minute. Add the spice mixture and cook, stirring, for 1 minute. Add the chicken stock, tomatoes, thyme and bay leaf, and bring to a boil. Return the beef to the pan, cover and transfer to the oven to cook until tender, about 45 minutes.
When done, remove the beef to a plate and cover to keep warm. Using potholders, move the pan to the stovetop and set over high heat. Cook, stirring occasionally, until the sauce has thickened into a gravy, about10 minutes. Season with a splash of vinegar. Add honey, salt and pepper to taste; stir in the parsley. Before serving, return the grillades to the pan, cover and warm them up over medium-low heat.
For the grits: Heat the butter and canola oil in a large saucepan over medium-high heat. Add the onion and cook until soft. Add the corn, season with salt and pepper and cook for a few minutes. Add the milk and 3 cups water and bring to a boil. Slowly whisk in the grits until well combined and continue whisking until smooth. Reduce the heat to low and cook, stirring occasionally, until thickened and smooth, about 45 minutes; add more water as needed to keep the mixture creamy. Remove from the heat, and whisk in the cheddar and Parmesan and the green onions.
To serve: Use a slotted spoon to transfer the grillades to a shallow platter. Pour over some of the gravy. Top with the grits, poached eggs if using, and some of the remaining gravy. Garnish with green onions and parsley.