Combine 1 cup lukewarm water, warm milk, yeast and sugar in a bowl. Stand until foamy, about 5 minutes
Meanwhile make flax egg.
In a large bowl combine flour and salt. Add butter and rub into flour between your fingers, making crumbs.
Stir in the yeast mixture and flax egg until a dough forms.
Pour dough onto a clean, un-floured surface and knead until smooth and elastic, About 8 to 10 minutes.
Shape dough into a ball and place it in a clean, lightly greased bowl.
Cover bowl with plastic wrap and let the dough rise in a warm place until doubled in bulk. I left mine for about 1.5 hours.
Line a baking sheet with parchment paper.
Divide dough into 8 equal parts. Gently roll each into a ball and arrange 2 to 3 inches apart on baking sheet.
Cover loosely with a clean kitchen towel and let buns rise in a warm place for around 1 hour.
Set a large shallow pan of water on oven floor. Preheat oven to 205 degrees C (400 F) with the rack in centre of the oven.
Combine 1 teaspoon almond milk with 1 teaspoon agave and brush on to buns. Bake, turning sheet halfway through baking, until tops are golden brown about 15 minutes.
Transfer to a rack to cool completely.
Combine the mayonnaise and kimchi juice in a bowl and set aside.
For the mushrooms, put the flour on a plate, the milk in a soup-size bowl and the Panko crumbs on another plate. Sit them beside each other with a large plate at the end to put the finished crumbed mushrooms.
Roll each slice of the mushroom in the flour, then the milk and finally roll in the crumbs.
Heat enough oil to fill a fry pan 1/2 inch.
Test heat by flicking some cold water in, if it sizzles, it's ready. Shallow fry the mushrooms for around 2 minutes on each side or until golden. Place on kitchen paper to drain.
To assemble the burgers, slice the buns in half and add a good dollop of kimchi mayonnaise to the bottom half. Add the crumbed mushrooms, followed by as much kimchi as you like and a handful of sliced cucumber.
Add the bun lid and serve.