Heat the olive oil in a medium skillet over medium-high heat. Add the onions and cook until translucent, about 3 minutes. Add the chanterelles, shiitakes and garlic and cook on medium heat until the mushrooms are fully cooked, about 5 minutes. Add the thyme and chile de arbol and mix well until evenly incorporated. Season with salt and pepper. Remove the mushrooms to a plate.
To the same skillet, add the white wine and bring to a simmer. Toss the Monterey Jack and Oaxaca cheese with the cornstarch until well coated. Slowly stir the cheese into the skillet and continue to gently stir until melted and smooth.
Pour into a fondue pot and keep warm. Top with the reserved mushrooms and serve with crostini or your favorite crudites.