Remove the lid from the Instant and select the Saute setting, when the Instant Pot is hot, add two tablespoons of extra virgin olive oil. 

When the oil is hot, add the onions and garlic to the Instant Pot, keep stirring the onions and garlic, so they don't become scorched or browned; continue to saute until the onions are soft and translucent, in about 8-10 minutes. 

Add the unsalted butter, chicken stock, and fresh sprigs of rosemary. Switch to the Steam setting, then pour and stir in the heavy whipping cream. Place the lid on the Instant Pot and allow the soup to steep for 10-minutes. 

After 10-minutes, crack and divide the egg yolks into a small bowl. Remove the lid from the Instant Pot and ladle 1/4 cup of the hot liquid into the egg yolks continuously stirring them and tempering them a little at a time. Add more of the hot liquid until the eggs are hot but have not curdled.

Slowly pour the egg mixture into the Instant Pot until the egg yolks are fully incorporated into the rest of the soup mixture. Continue to cook for five additional minutes.

Place a large fine sieve over a large bowl and pour the soup through the sieve, strain out the sprigs of rosemary, onions, and garlic and discard. Pour the strained soup back into the Instant Pot. 

Select the Saute setting and add one cup of grated Parmigiano-Reggiano, and continue to stir with a wire whisk or blend with a wand mixer until the cheese is melted thoroughly into the soup. 

Turn off the Instant Pot and allow the soup to set for 10-minutes. 

Serve with buttered, toasted bread with melted cheese and fresh rosemary sprigs on top.