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For matcha noodles, combine flours, matcha and ½ tsp fine salt in a large bowl and mix to combine.
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Add 150ml water and mix with your fingertips. Gradually add another 150ml water, a little at a time, mixing until you have a soft but not sticky, dough (you may not need all the water).
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Turn out onto a surface lightly dusted with buckwheat flour and knead until smooth (4-5 minutes).
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Divide into 4 pieces and form each into a round. Roll each to a 2mm-thick rectangle, dusting generously with buckwheat flour as you go to prevent sticking.
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Lightly dust top of dough to prevent sticking, then fold in half and cut into 2mm-wide noodles with a very sharp knife. Or put through the tagliarini cutter on a pasta machine. Carefully unfold noodles and toss in buckwheat flour to prevent sticking.
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Cook eggs in a saucepan of boiling water for 7 minutes (for soft yolks), drain and run under cold water until cool enough to handle.
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Meanwhile, bring tamari, ginger and 1.5 litres water to a simmer in a saucepan. Whisk 250ml of the hot ginger liquid in a bowl with miso to combine, return to pan and simmer for flavours to combine (4-5 minutes; do not boil). Add mushrooms and spinach and simmer until spinach is just wilted (30 seconds to 1 minute).
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Blanch noodles in a large saucepan of boiling salted water until shiny and tender (30 seconds to 1 minute). Drain and divide among warmed serving bowls, then ladle broth, spinach and mushrooms over noodles. Crack eggs in half, scoop out of shells and add to bowls.
