Cook ¾ orzo to al dente, according to package directions. Drain and add orzo to a large mixing bowl.
Heat 1 tablespoon oil in a skillet over medium until shimmering. Add ½ cup minced shallot and cook until softened, about 2 minutes.
Stir in 2⅓ cups shredded carrots and 4 ounces arugula; cook until vegetables are wilted, about 2 minutes.
Stir in 2 minced cloves garlic, 1 tablespoon curry powder, and 1 teaspoon salt, cook until fragrant, 2 minutes; season with pepper to taste. Add vegetable mixture to bowl with orzo; toss to combine. Wipe out skillet.
Add ¼ cup yogurt, 2 eggs, and ¼ minced cilantro to orzo-vegetable mixture and stir to combine. Scoop mixture by ¼ cup and pack into ½-inch-thick cakes. Transfer cakes to parchment-line sheet and chill until firm, about 30 minutes or up to overnight.
Heat 1 tablespoon each oil and butter in skillet over medium until shimmering. Gently lay 4–5 cakes in skillet and cook until golden brown and crisp on both sides, about 3–5 minutes per side. Transfer cakes to a paper-towel-lined plate. Repeat with remaining cakes.
Serve cakes with a dollop of Greek yogurt and scallion greens.