Heat oven to 350°F (175°C). Grease and line a 9×13-inch (23×33 cm) baking pan with parchment paper.
In a small bowl, stir together ground flaxseed and ½ cup of the plant milk. Let sit for 5 minutes to thicken.
Prepare the cinnamon swirl: In a separate small bowl, mix brown sugar and cinnamon. Set aside.
Make the crumb topping: In a medium bowl, combine flour, both sugars, and cinnamon. Cut in the cold vegan butter with a fork or pastry cutter until the mixture is crumbly. Set aside.
Make the cake batter: In a large bowl, whisk together melted vegan butter, brown sugar, thickened flax mixture, vegan Greek yogurt, and vanilla extract until smooth. Add flour, baking powder, and salt. Stir gently until just combined. Pour in the remaining ½ cup of plant milk and mix until the batter is smooth and thick. Do not overmix.
Assemble the cake: Spread half of the batter evenly in the prepared pan. Sprinkle all of the cinnamon sugar mixture over the first layer. Gently spread the remaining batter on top. Evenly sprinkle the crumb topping over the surface.
Bake for 45–50 minutes, or until a toothpick inserted in the center comes out clean.
Allow the cake to cool in the pan for at least 20 minutes before slicing. Serve warm or at room temperature with coffee or tea.
Notes
Room temperature: Keep covered in the baking dish at room temperature for up to 3 days, or transfer leftovers to an airtight container.
Freezer: Wrap individual slices in plastic wrap and freeze for up to 3 months in a zip-top freezer bag or airtight container. Thaw at room temperature or in the microwave before serving.