Pour the stock into a 2 L / 2-quart saucepan and bring to a simmer over medium heat. Lower the heat to keep it barely bubbling while you cook the risotto.
Heat the olive oil and 30 g butter in a 26 cm / 10-inch heavy-bottomed saucepan over medium heat. Add the shallot and cook for 3-4 minutes, stirring, until soft and translucent but not browned.
Add the rice and stir constantly for 2 minutes until the edges look glassy and every grain is coated in fat. Do not let it brown.
Pour in the Prosecco. Let it bubble and mostly evaporate, stirring, about 2 minutes, until the sharp alcohol smell fades and only a faint fruity note remains.
Add a ladle of hot stock, stirring until it's nearly absorbed, then add the next. Repeat for about 17-18 minutes, keeping the pot at a steady simmer, until the rice is al dente with a firm bite in the center and looks loose and creamy around it.
Season lightly with salt partway through, keeping in mind the Parmigiano added later is already salty.
Remove the pot from the heat and let it rest, uncovered, for 2 minutes.
Add the cold cubed butter and grated Parmigiano. Beat vigorously with a wooden spoon until the risotto turns glossy and moves in a slow wave when you tilt the pot.
Taste and adjust salt and black pepper. Spoon onto warm plates right away, since risotto keeps cooking and tightens up as it sits.
Garnish with chives and extra Parmigiano if using. Serve immediately.