In a large skillet over medium heat, heat oil. Add sausage and cook, stirring occasionally, until golden brown on both sides, 6 to 8 minutes. Transfer to a plate.
In same skillet over medium-high heat, cook onion, stirring occasionally, until softened, about 7 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more. Add orzo and stir to toast, about 1 minute. Add wine, reduce heat to medium, and bring to a simmer. Cook, stirring occasionally, until liquid is completely reduced, 1 to 2 minutes. Pour in broth, then stir in cream, pumpkin, sage, thyme, and cayenne; season with salt and black pepper. Bring to a simmer, then reduce heat to low. Cook, stirring often, until orzo is al dente, about 10 minutes.
Return sausage and any accumulated juices to skillet. Stir in cheese until melted and combined; season with salt and black pepper, if needed.
Divide orzo mixture among plates. Top with sage and thyme.