"That Old Feeling" was written by Sammy Fain, with lyrics by Lew Brown. The song was published in 1937. Frank Sinatra did a sublime version on his 1960 "not too happy, not too sad/not too fast, not too slow" album, Nice 'n' Easy. It has some great examples of the singer's incredible breath control (the "And when you caught my eye, My heart stood still" section - WOW! That's lung power) and clear pronunciation of the lyric. And of course, it's Sinatra, so he's singing to you, and only you:
On a side note, the song first appeared in the 1937 movie Vogues of 1938.