Preheat oven to 350°. Butter a 10" springform pan and line bottom with a round of parchment paper. Butter paper, then coat pan and paper with Parmesan, tapping out excess. In a large pot of boiling salted water, cook cauliflower over medium-high heat until soft, about 7 minutes. Drain well and spread out on a baking sheet to cool and dry.
In a large skillet, heat olive oil. Add chopped onion and rosemary and season with 1/2 tsp. each salt and pepper. Cover and cook over medium-low heat, stirring occasionally, until softened, about 8 minutes. Transfer to a large bowl and let cool completely.
Meanwhile, in a large bowl, whisk together flour, baking powder, turmeric, 2 tsp. salt, and 1 tsp. pepper. Stir in Parmesan. Stir eggs and basil into onion mixture, then add to flour mixture. Fold in reserved cauliflower. Scrape mixture into prepared pan and top with onion rings. Bake until golden brown on top and set in the center, about 40 minutes. Remove cake from oven and let cool in pan 10 minutes. Remove pan ring, cut cake into wedges, and serve warm or at room temperature.