• Preheat the oven to 180°C (356°F). Grease the base of a 7"/18-cm square cake pan, line with parchment paper, and grease again.
• Bake the walnuts for 10 minutes until lightly toasted. When cool enough to handle, finely chop into small bits. Set aside 50 grams for the cake, and reserve the rest for decorating.
• Sift together plain flour, baking powder, and salt into the mixer bowl. Add the sugar and start the mixer fitted with a paddle on low speed. Mix for 1 minute.
• With the paddle still turning, add the softened butter, 2 - 3 cubes at a time. Stop the mixer when the mixture resembles coarse breadcrumbs with some pea-size clumps.
• Add the milk and vanilla extract all at once. Mix at low speed for 30 seconds, then increase to medium speed and beat for 30 seconds. Scrape the bottom and sides of the mixer bowl.
• Lightly beat the eggs with a fork. With the mixer turning at low speed, add 1 egg. Increase the speed to medium and beat for 30 seconds. Scrape the bottom and sides of the bowl.
• Repeat with the second egg, scraping the bottom and sides of the bowl after each addition until all 3 eggs are beaten in.
• Scrape the bowl once more, then beat the mixture at medium-high speed for 30 more seconds. Reduce the mixer speed to low, and tip in 50 grams of chopped walnuts (reserve the rest for decorating). Mix for 15 seconds until the walnuts are well distributed throughout the batter.
• Spoon out the batter into the greased pan, spread evenly and smoothen the surface with an offset spatula. Bake on the middle rack in the preheated oven for 40 minutes, or until the cake is risen, golden brown, and firm. The cake is done when the edges start to pull away from the sides of the pan, and a metal skewer inserted into the center of the cake comes out clean. 
• Remove the cake from the oven and let it sit in the pan for 2 - 3 minutes. Then turn it out onto a wire rack and peel off the parchment paper. Leave to cool completely.

Make the coffee buttercream

• In a small heatproof bowl, combine egg whites and sugar and whisk to mix well. Place the bowl over a saucepan or pot filled with 1 -2" (2.5 - 5 cm) of simmering water. Make sure the base of the bowl sits above the water level. Keep whisking until all the sugar dissolves and no longer feels grainy between your fingers, or until the temperature reaches 71°C (160°F).
• Transfer to a stand mixer bowl fitted with a whisk and whip on medium-high speed until the meringue is stiff and no longer warm to the touch, about 10 minutes.
• Switch out the whisk for a paddle. Beat in the butter cubes, 2 to 3 pieces at a time, at medium speed. The mixture will start to curdle and look awful, but just keep going!
• After incorporating all the butter, keep beating until the buttercream becomes light, smooth and fluffy. This may take several minutes.
• Reduce the mixer speed to low. Pour in the coffee solution and beat for 10 seconds until well mixed. Set aside in a cool, dry place.

Assemble the cake

• Place the apricot glaze in a heatproof bowl and microwave on high for 15 - 20 seconds. Spread over the top of the cake, leaving about 0.5"/1-cm of the edges free.
• Using an offset spatula, frost the sides of the cake with the coffee buttercream. Gently press the remaining chopped walnuts into the buttercream with your palm or a scraper.
• Fit a piping bag with your choice of piping tip and fill with the remaining coffee buttercream. Pipe the desired patterns over the apricot glaze and around the edges of the cake. Decorate with whole walnuts on the piped buttercream.
• Store the cake stored in a cool, dry place like a turned off, cool oven at room temperature, until ready to slice and serve. The cake will keep well in a sealed and air-tight container for 3 - 5 days.