Bring water, sorghum, 1 teaspoon oil, and ½ teaspoon salt to a boil in a medium saucepan over high heat. Reduce heat to low, cover, and simmer 80-85 minutes until grains are tender and water has been absorbed. Remove from heat, fluff with a fork, and let cool completely.

Line a baking sheet with parchment paper.

Transfer cooked sorghum to a food processor and pulse until the grains are chopped and start to stick together, about 20 one-second pulses; transfer to a large bowl.

Heat 1 tablespoon oil in a skillet over medium until shimmering. Add shallot and cook until softened, about 2 minutes. Stir in arugula and carrots and cook until arugula is wilted, about 2 minutes. Stir in garlic, curry powder, and 1 teaspoon salt, cook until fragrant, about 30 seconds; season with pepper. Add arugula mixture to bowl with sorghum and toss to combine; wipe out empty skillet.

Add ¼ cup yogurt, egg, and cilantro to sorghum-arugula mixture and stir to combine. Divide mixture into 8 portions (¼ cup each) and pack into firm ½-inch-thick cakes. Transfer cakes to prepared sheet and chill until firm, about 30 minutes or up to overnight.

To bake the sorghum cakes:

Heat oven to 350°F. Coat a baking sheet with nonstick spray then transfer chilled cakes to sheet. Coat tops of cakes with nonstick spray and bake 15 minutes. Flip cakes and bake 10–15 minutes more until edges are crisp and cakes are cooked through.

To fry the sorghum cakes:

Heat oven to 200°F and set wire rack inside baking sheet.

Heat 1 tablespoon oil in skillet over medium until shimmering. Gently lay 4–6 cakes in skillet and cook until golden brown and crisp on both sides, about 5 minutes per side. Transfer cooked cakes to prepared sheet and keep warm in the oven. Repeat with remaining cakes and additional oil.

Serve cakes with a dollop of lightly sweetened greek yogurt, cilantro, and scallion greens.