Preheat the oven to 400 degrees F. Preheat the oil in a deep fryer.
Combine 1/4 cup of the vegetable oil and flour in a saucepan over medium heat. Stir the mixture constantly 5 to 6 minutes for a blond roux. Whisk in the milk and bring the liquid up to a boil. Season the liquid with salt and cayenne. Simmer the liquid for 5 to 6 minutes, or until the liquid is thick and coats the back of a spoon. Remove the sauce from the heat and stir in the cheeses. Set the sauce aside.
In a saute pan, heat the remaining 2 tablespoons vegetable oil. When the oil is hot, add the onions and saute for 2 minutes. Stir in handfuls of spinach at a time, until all the spinach is incorporated. Add the garlic and artichoke and saute for 2 minutes. Season the vegetables with salt and cayenne. Remove the vegetables from the heat and turn into a mixing bowl. Fold the cheese sauce into the vegetables. Turn the mixture into a baking pan. Bake the dip for 10 to 15 minutes, or until the top is golden brown.
Fry the tortilla chips, batches at a time, for about 2 to 3 minutes or until the chips are golden and crispy. Remove from the oil and drain on a paper-lined plate. Season with salt and pepper. Serve the chips with the dip.