Melt the butter in a medium pot over medium-high heat. Add the pasta and cook until nutty and deep golden brown.
Add the EVOO, 1 turn of the pan, then the asparagus, carrot, shallots and garlic; season with salt and pepper. Cover and cook to sweat the vegetables a bit, stirring occasionally, for 5 minutes. Stir in the stock, rice, lemon zest, thyme and browned pasta and bring to a boil. Cover the pot, lower the heat and simmer until the rice is just tender, 16 to 18 minutes. Fold in the arugula, then stir in the lemon juice and cheese.