Preheat the oven to 425 degrees F. Combine 1/2 cup water, 2 tablespoons salt, 1 tablespoon brown sugar and 4 sage leaves in a small saucepan over medium-high heat, stirring, until the sugar and salt dissolve. Pour the mixture into a large bowl and add 5 cups cold water. Submerge the pork chops in the brine and refrigerate 10 minutes.
Chop 3 sage leaves; set aside. Toss the remaining 5 sage leaves with the squash, 2 tablespoons olive oil, the nutmeg and 1/2 teaspoon salt. Spread on a baking sheet and roast until tender, 25 to 30 minutes.
Heat the remaining 1 tablespoon olive oil in a large nonstick skillet over medium-high heat. Drain the pork and pat dry. Add to the skillet and cook until a meat thermometer inserted sideways into the center registers 140 degrees F, 5 to 7 minutes per side. Divide among 4 plates.
Add the chicken broth and the remaining 1 tablespoon brown sugar to the skillet; simmer, stirring, until reduced by half, about 3 minutes. Remove from the heat and add the lemon juice, the reserved chopped sage, and salt to taste. Drizzle the pork with the sauce and serve with the squash.