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Preheat the oven to 400°F. On a large rimmed baking sheet, toss together the squash, garlic, sage, chile, 2 tablespoons of the olive oil, the maple syrup, and a pinch of salt. Bake until the squash is lightly browned and caramelized, about 40 minutes.
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Heat a large ovenproof sauté pan over medium high heat. Add the remaining 1 tablespoon olive oil. Season the pork chops with salt and pepper on both sides, add them to the hot oil, and cook until golden brown, about 3 minutes per side. Transfer the pan to the oven and cook until the internal temperature of the chops reaches 160°F, 6 to 7 minutes.
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Meanwhile, in a small saucepan over medium heat, cook the butter until it begins to darken and smell of toasted nuts. Add the vinegar and immediately remove the pan from the heat.
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Transfer the roasted squash mixture to a platter, top with the pork chops, drizzle with the brown butter vinegar, garnish with cilantro, and serve.