Preheat oven to 350 degrees Fahrenheit.
Cut the mushroom in half. Warm an 8-inch, ovenproof skillet over medium heat, then add just enough oil to coat its surface. Add mushroom, flat side down, and drizzle it with 1 teaspoon oil, then season with salt.
Move skillet to oven and cook for 3 minutes.
Meanwhile, heat a second, cast-iron skillet on medium just until the pan starts to smoke. Add Oaxaca cheese evenly, and do not touch.
Turn heat down to medium low. When you notice cheese browning at the edges, remove skillet from heat and place in the oven.
Let the cheese melt completely, then remove skillet from oven.
Remove the other skillet from the oven. Flip over the mushroom, then add butter and garlic to the still-hot skillet around the mushroom. When butter has melted and garlic is fragrant, transfer the mushroom, garlic, and butter to center of cheese.
Season with salt and chives. Serve immediately, while cheese is still bubbling, with hot flour tortillas.