Mix the warm milk, yeast and half the granulated sugar. Leave for 5–10 minutes, until foamy. The milk should feel gently warm, not hot.
Mix the flour, remaining sugar and salt. Add the yeast mixture, melted vegan butter, vanilla and oil.
Mix into a dough. It should feel soft and slightly tacky, not dry. If it will not come together, add soy milk ½ teaspoon at a time. Avoid adding extra flour unless it is genuinely unmanageable.
Cover and let rise until roughly doubled, about 60–90 minutes in a warm room.
Lightly flour the work surface. Roll dough into a 3 x 17 inch (8 x 45 cm) rectangle.
Spread the softened butter on the dough.
Make the cinnamon sugar by mixing cinnamon and brown sugar. Sprinkle over the butter.
Cut dough in half lengthwise. Roll up from one 1.5-inch edge, forming one spiral.
Place spiral over the second strip of dough and roll up to form a larger spiral. Place spiral-side up in oiled ramekin (3 x 3 inch - 7.5 x 7.5cm; 7 oz - 200 ml)
Cover and let rise for 60 minutes, until noticeably puffy and expanded.
Preheat air fryer to 300°F (150°C) for about 3 minutes. Bake at 300°F (150°C) for 11 minutes.
Let the roll cool down, then add the icing.