Start by roasting the poblano chiles over an open flame or in the oven or air fryer until the skin is charred and blistered. Place them in a bowl and cover with a clean kitchen towel for 10 minutes to steam. Once cooled, peel off the charred skin, remove the seeds, and dice the chiles. Set aside.

In a large pot, heat the vegetable oil over medium heat. Add the diced onion and cook until translucent, about 5 minutes. Stir in the minced garlic and cook for another minute, being careful not to burn it.

Toss in the diced tomatoes and cook for 3-4 minutes until they start to break down. Pour in the water or vegetable broth and bring to a gentle simmer.

Add the diced potatoes to the pot. Cover and let them cook over medium heat for about 15-20 minutes, or until they are tender but not falling apart.

Stir in the roasted poblano chiles and warm milk. Season with salt, pepper, and oregano (if using). Simmer for another 5 minutes to let the flavors meld together. Slowly add the cheese and let it melt.

Serve warm with freshly made corn tortillas.