2 cups very thinly sliced assorted summer squash (such as zucchini, yellow crookneck, and pattypan)
1/4 cup thinly sliced shallots
1/4 cup thinly sliced fresh basil, divided, plus 1/4 cup basil leaves
20 cherry tomatoes, halved
4 tablespoons dry white wine
4 tablespoons extra-virgin olive oil, divided
Kosher salt, freshly ground pepper
4 6-ounce skinless white flaky fish fillets (such as Atlantic cod or halibut)