Preheat the oven to 375 degrees F (190 degrees C).
Brush oil on both sides of tortillas.
Invert ramekin on a work surface. Center tortilla over bottom of the ramekin; gently shape tortilla down around sides.
Place a double layer of aluminum foil over tortilla and press to form a bowl around the ramekin.
Place right-side-up on a baking sheet. Leave about 1/2-inch of tortilla exposed above foil, so it gets browned and crispy when baked. Repeat this step with second tortilla.
Bake in the preheated oven for 5 minutes. Use tongs to remove the ramekins from shells. Continue baking shells until lightly browned, about 5 more minutes.
Carefully remove foil from around shells. Flatten foil sheets and invert shells on them. Bring foil up just around bottom of shells to protect top edge.
Bake shells until browned and crispy, 8 to 10 minutes. Invert shells and continue baking until crisp, 3 to 5 minutes. Watch them carefully, so they don't burn.
Transfer basket burritos to a rack to cool completely.