Place the mackerel fillets in a small container at room temperature. Pour sake on both sides of the fillets and let rest in the container, covered, for 30 min.
Cut onion into large slices.
To a saucepan, add two cups of water, sliced onions, and miso paste. Whisk the solution until the miso paste is fully dissolved.
Bring to a boil. Let the miso soup boil for about 10-15 min, then bring to a simmer.
While the miso boils, peel the summer squash. Cut squash into 1/4 in half moons.
In a large pan, melt 1 tbsp butter at medium-high heat.
Grill squash in the pan for 5-10 min, until both sides of the squash are golden brown. Add salt and pepper to taste.
Push squash one side of the pan.
Season each fillet with about 1 tsp of salt.
Place the fillets in the pan with the squash. Grill each side of the mackerel for about 4 min.