Peel the first 1-2 layers of the leek and cut the top green part of the leek. Wash well to remove any dirt.
Slice the root ends of the leek then cut into 1-inch/2.5 cm thick pieces.
For the king oyster mushrooms, slice horizontally in half. Lightly score diagonally on the surface and repeat at an angle to create a cross-hatch pattern. Do not cut through the mushrooms.
Season the scored side of the mushrooms with salt and pepper.
Heat a medium or large skillet or cast iron pan over medium heat. Add enough oil to lightly coat the surface of the pan.
Once the oil is hot (you can see ripples in the oil), add the mushrooms scored side down. Allow the mushrooms to sear over medium heat until golden brown. Flip and cook on the remaining side until golden brown.
Remove the mushrooms from the pan and place on paper towels to remove excess oil.
In the same pan with oil, place the sliced leeks. Sprinkle some salt and pepper over the leeks. Cook until the leeks turn golden brown then flip to repeat for the remaining side.
Meanwhile, mix together softened vegan butter and miso paste until combined.
Add the mushrooms back into the pan with seared leeks. Add the garlic, butter-miso mixture, and fresh thyme. Melt the butter-miso mixture and tilt the pan towards you. Using a spoon, scoop the butter mixture and pour over the leeks and mushrooms to continue cooking and infuse flavor. Repeat for 1-2 minutes or until the mushrooms and leeks are coated in the butter-miso mixture and cook down more.
Lower the heat. Add the mirin and shift the leeks & mushrooms around to coat in the mirin.
Turn off the heat. Finish with lemon juice.
Serve hot with more thyme for garnishing, if desired.