We heat a pan, in which we will cook the vegetables, over low heat. We add the olive oil and the onion to the pan and begin to slowly cook the onion.
Meanwhile, we rinse our zucchini and pat it dry. We cut off and discard both ends, and then slice it half, lengthwise. Each half is cut in quarters lengthwise and then all are diced. We end up with 2¼ cups of zucchini. The onions are translucent, so we raise the heat to medium and add the zucchini. We give the vegetables a quick stir.
In a large measuring cup, we add the frozen corn. We cook in the microwave on normal heat for a minute, giving them a quick thaw.
We stir the zucchini again—they are cooking nicely but are still not tender. We smash the garlic clove with the side of our knife (carefully) and discard the peel. We sprinkle with coarse salt and mince.
We add the butter to a pot, put the pot over low heat, and wait for the butter to melt.
When the zucchini have been cooking for about 5 minutes and are tender, we stir in the corn and garlic. We cook for a minute, stirring occasionally, and lower the heat. We open the can of green chiles and add them, including the liquid, to the pan.
We add the flour to the just-melted butter, and stir with our flat whisk.
The large measuring cup is given a quick rinse and 2 cups of the milk are added. We heat the milk in the microwave for a minute.
We add the chile powder and cumin to the pot, and stir well.
When the milk is warm, we add it gradually to the pot, whisking furiously. When the entire 2 cups has been added, we add the remaining ½ cup (cold, from the fridge).
At this point, the vegetable mixture is ready. We taste for seasoning and add some salt and freshly ground black pepper and then take the pan off the heat.
As the sauce in the pot is brought to a boil, we grate the cheese with our box grater. We want ¾ cup of each of the cheeses. The rest of the cheese is wrapped and refrigerated for another use.
When the sauce is boiling, we whisk some more and lower the heat. We cook for a couple minutes more and then turn the heat off. Most of the cheese is stirred in, but ½ cup has been reserved for the top of the assembled dish. We season the sauce with salt and pepper and then mix ¾ cup of the sauce in with the cooked vegetable mixture.
On a dinner plate, we wrap the corn and flour tortillas in a damp paper towel and heat in the microwave for a minute to soften them.
We coat the bottom of our 9 × 13 inch glass baking dish with a little of the cheese sauce. We are now ready to fill the tortillas.
We take one corn tortilla (keeping the rest covered with the paper towel) and put about a quarter cup of the vegetable-cheese mixture down the center. We roll it up, and place it in the dish seam-side down. We repeat with the remaining corn tortillas, placing them close together in the pan. The seven corn tortillas fit crosswise in the baking dish.
We divide the remaining filling between the two flour tortillas. We place them lengthwise to completely fill the pan.
A little of the vegetable mixture remains, which we spread on top of the rolled tortillas. The rest of the sauce is also spread on top of the tortillas. Finally, the reserved cheese is sprinkled on.
We place the baking dish, uncovered, in the preheated oven and cook for 30 minutes.
The bubbling casserole is removed from the oven with potholders and placed on a trivet. The cilantro is sprinkled on top and then the pan is covered with foil. We let it rest for 5 minutes to allow the enchiladas to set up and then serve.