Teezy

Sometimes when Daddy comes home from work, he makes a "teezy". He always seems to want more of this drink since he keeps saying he's making "more" "teezy". This is a fun drink to make. First, he puts a wet glass in the freezer to cool. Then he puts some ice cubes and some juices called gin and vermouth into a shaker. Then the fun begins. He shakes the shaker and it makes a great maraca sound. Then he pours the "more" "teezy" into his glass. Then it's my turn to shake the shake. I used to only give it a desultory shake and then return it to Daddy saying "Daddy's turn". But now I shake with gusto all the while chanting "teezy, teezy, teezy". Then I pass it back to Daddy for more shaking. When I want it back, I request "More teezy"

Can you guess what this drink is?