Whisk 3 cups flour, granulated sugar, yeast, and salt together. Add warm almond milk, oil, and egg, and stir to a shaggy dough. Knead 5 minutes on a floured surface until smooth and elastic, adding up to 1/4 cup more flour as needed.

Place dough in an oiled bowl, cover, and let rise in a warm, turned-off oven until doubled, 45 minutes to 1 hour.

Punch dough down and roll into a 12x18-inch rectangle. Brush with melted coconut oil, leaving a 1/2 inch border on one long side. Sprinkle brown sugar and cinnamon evenly over the oiled dough and roll up tightly from the long side toward the border.

Cut into 9 even rolls with a serrated knife or dental floss, place cut side up in a greased 9x13-inch pan, cover, and let rise until puffy and touching, 30 to 45 minutes.

Preheat oven to 350°F (175°C). Bake rolls 20 to 25 minutes until golden. Whisk powdered sugar, 2 tablespoons almond milk, and vanilla into a smooth icing, adding the last tablespoon of milk if needed. Cool rolls 5 minutes, then drizzle with icing and serve warm.