In a large stainless steel bowl, add yeast to warm water and allow time for yeast to bloom (becomes frothy and smells yeasty), about 10 minutes. In a second bowl, whisk eggs with milk. In a third bowl, cream together sugar, butter, lemon zest and vanilla.
Add egg/milk mixture to sugar/butter and incorporate. Add this to yeast mixture and combine. Add 2 c. of flour and mix together; add last 2 c. of flour, being careful not to add all at once as batter should be slightly sticky.
Place batter in a bowl, cover with plastic and chill for at least 2 hours.
Remove batter from bowl and place on lightly floured surface and roll out to 1/2-in. thick. Heat oil in a stock pot on stove to 350°F/180°C.
Cut batter into 3-in. squares and fry in batches in hot oil. Once batter is dropped in, beignets should float to top. Flip over and continue to fry until light brown all around. Remove to absorbent paper and dust with confectioners' sugar.