Remove tips from asperagus and slice stalks into thin disks.
Pour stock and water into medium saucepan and heat to low simmer. In 3-quart or larger pot, heat olive oil and 1 tablespoon butter over medium heat. Add onion and saute until soft and translucent. Add rice and cook, stirring constantly, until all the grains are well-coated. Quickly pour in vermouth or wine and stir until it has evaporated.
Turn heat down to medium low and begin adding warmed stock to rice a ladleful (about 1/2 cup) at a time, stirring after each addition and waiting until the stock has been absorbed before adding more. Stir often enough to prevent rice from sticking to pan.
When about half the stock has been absorbed, stir in asparagus slices and tips, then continue adding stock by the ladleful and stirring until it is absorbed. About 20 minutes after you began the process, taste rice for doneness. It should be tender, but not mushy, with a bit of crunch at the center. You may not need to add all the stock. The end texture should be creamy and slightly loose.
Turn off heat and stir in remaining butter, cheese, mint and lemon zest and juice. Check seasoning and add more salt and pepper if needed. Cover pan and let risotto rest for a few minutes before serving. Pass additional Parmesan at the table.