Pour stock into a saucepan and bring to a boil. Reduce heat to keep hot.
Meanwhile, in a large saute pan, cook the onion in 3 Tbsp. of olive oil until translucent and tender. Season with salt and pepper. Add the orzo and mix thoroughly. Gradually add the stock to the pasta, one ladle full at a time, stirring each continuously, until the stock is absorbed and then adding another ladle full. Continue this process until the orzo is cooked to al dente.
While the orzo is cooking, in a separate saute pan, heat 2 Tbsp. of olive oil and sear the mushrooms and garlic scapes until golden brown. Add 2 Tbsp. of the butter and a pinch of salt and freshly ground pepper. Let the butter become golden brown, then remove from heat and reserve.
Once the orzo is cooked, stir in the remaining 2 Tbsp. of butter, Parmesan cheese and heavy cream. Stir to combine and then stir in the baby arugula and cook just until the arugula is wilted. Taste and add additional salt and pepper, if necessary. Return the mushrooms to a hot burner briefly to re-warm.
To serve, spoon orzo into a shallow bowl and place the a spoonful of the mushrooms and scapes over the top. Using a vegetable peeler, shave some additional Parmesan cheese on to the top for garnish.