Heat the avocado oil over medium-high heat in a large skillet. Add the onion and sauté, stirring occasionally, until the onion is translucent, about 5 to 8 minutes.
Stir in the mushrooms and garlic and cover the skillet with a lid. Cook, stirring occasionally, until the mushrooms have leached out much of their water, about 8 minutes. Remove the lid and continue cooking until most of the liquid has evaporated.
Pour in the canned coconut milk, dried oregano, nutmeg, and sea salt and stir well. Bring the mixture to a full boil, then add the flour and whisk well.
Continue cooking and whisking, leaving the mixture at a full boil, until it thickens to the same consistency as a very creamy soup, about 10 to 15 minutes. If needed, add 1 to 2 tablespoons of additional gluten-free flour if the soup isn't becoming thick enough.
Remove the cream of mushroom soup from the heat and set aside to allow it to cool.
Once cool, either use the soup in a recipe, or transfer it to a jar. Seal and refrigerate for up to 1 week.