Preheat the oven to 160 °C (fan). Line two 20cm/7.8 inch spring-form cake tins with greaseproof paper. Set aside.
Place the walnut pieces, caster sugar and light brown sugar in a food processor and process until sandy textured with no chunks of walnut remaining.
60 g walnut halves, 180 g caster sugar, 45 g light brown sugar
To the food processor, add the vegetable oil, vegan yoghurt, soy milk and instant espresso powder then run again until the mixture is thick and smooth. Set aside.
200 g vegetable oil, 200 g vegan yoghurt, 4 teaspoon instant espresso powder, 4 tablespoon soy milk
In a medium bowl, sieve together the plain white flour, corn starch, baking powder and bicarbonate of soda. Add the wet ingredients to the sieved dry ingredients and mix until just combined - the mixture will be a thick batter.
220 g plain white flour, 2 tablespoon corn starch, 2 ½ teaspoon baking powder, ½ teaspoon bicarbonate of soda
Divide the batter between the two prepared cake tins and smooth the tops of the batter with the back of a spoon. Bake the two cakes for 25 minutes.
Remove the cakes from the oven and turn the oven temperature to 140 °C. Leave the cakes to cool in their tins for 15 minutes. Remove the tins and transfer the cakes to a wire cooling rack to cool completely before transferring to the fridge to chill. While the cakes cool, make the candied walnuts, coffee buttercream and coffee syrup.
To Make the Candied Walnuts
Place the walnut halves in a sieve or colander and run under cold water. Toss until all walnuts are wet all over. Place the walnuts in a small bowl and add the granulated sugar. Toss to coat the walnuts in the sugar.
125 g walnut halves, 50 g granulated sugar
Spread the walnuts out on a baking tray lined with baking parchment. Place in the oven and bake for 40 minutes, stirring halfway through.
Once baked, remove from the oven and allow to cool fully on the baking tray.
To Make the Coffee Buttercream
Place the prepared batch vegan buttercream in a medium bowl. In a separate small bowl, combine the instant espresso powder and boiling water then leave to cool for 5 minutes.
Add the coffee mixture to the vegan buttercream and stir until combined. Set aside.
To Make the Coffee Syrup
In a medium bowl, whisk together the coffee syrup ingredients until the sugar and espresso powder are completely dissolved.
50 g caster sugar, 1 tablespoon instant espresso powder, 50 ml boiling water, ½ teaspoon vanilla extract
Remove the chilled coffee walnut sponges from the fridge and place upside down on a cooling rack. Divide the coffee syrup between the two sponges, drizzling it over and allowing to soak in fully before adding more.
To Assemble the Cake
Top one layer of sponge with a quarter of the coffee buttercream. Top with the other layer of sponge then use the remainder of the buttercream to frost the cake. Top the cake with 10 candied walnut halves one per slice) and push the remainder into the buttercream around the outside of the cake.
Serve the cake or place in a sealed container in the fridge for up to two days.