How to Make It
In a small saucepan, bring the vinegar, sugar, salt, sage and nutmeg just to a boil, stirring to dissolve the sugar. Add the chiles and let cool completely, then refrigerate until chilled.
In a saucepan, warm the honey, sage, nutmeg and a pinch of salt for 3 minutes. Let steep for 30 minutes.
Preheat the oven to 375°. On a large rimmed baking sheet, toss the squash with the butter and piment d’Espelette and season with salt and pepper. Roast for about 20 minutes, until just tender; let cool.
Spread 1/2 cup of the rice flour in a shallow bowl. In a large bowl, whisk the all-purpose flour with the cornstarch and remaining 3/4 cup of rice flour. Whisk in the sparkling water, vodka and a pinch of salt until the batter is smooth.
Line 2 large baking sheets with paper towels. In a large saucepan, heat 2 inches of oil over moderately high heat to 350°. Add the sage leaves and fry until crisp, about 1 minute. Using a slotted spoon, transfer the crispy sage to the paper towels to drain.
Working in 3 batches, dust the squash with the rice flour, then coat in the batter and fry until golden, 4 to 5 minutes per batch. Transfer to the paper towels to drain. Transfer the fried squash and sage leaves to a platter. Drizzle some of the sage honey on top and garnish with Parmigiano and some of the pickled chiles. Serve with lemon wedges, passing additional sage honey and pickled chiles at the table.