Looking for a butter thin cookie recipe

Sounds like you;re looking for a sort of langues de chat (spelled right? not sure) - literally cat's tongue.  I believe you take a small ball of the dough and roll it very hard so it elongates.  Bake on a baking sheet.  To get the caramelized characteristic, you would roll it in sugar first then roll out.  I have a recipe for a cookie that's similar. You make balls of the dough, roll in sugar and squash with a sugared bottom of a glass.  If you'd like it just ask.  But it;s not that thin. 

Langues de chat (langues du chat?) are pretty common and you can surely find a recipe - i never made them. 

Ok, you might try this, this is the one i make with the glass.  (I got it in a magazine, like better homes or good housekeeping - don;t scoff at the source, they have some good recipies!) - and it was intended to have white chocolate drizzled on top and chopped pistachios sprinkled on top of that so they stick on the melted chocolate.  But i think they;re good plain cookies as they are, the bit of sugar gets caramelized a little and they;re crisp and tasty.

Preheat oven to 350

1 egg

1 1/2 cups flour

3/4 cups butter

1/2 cups powdered sugar

1/2 tsp salt

granulated sugar to roll the dough in later. 

beat butter and powdered sugar till creamy.  Beat in egg, then beat in flour. 

Cover and chill one hour

Shape into 1/4 inch balls. Roll in granulated sugar.  put on ungreased cookie sheets. Dampen or grease the bottom of a glass or other flat thing and dip in granulated sugar.  press down the cookie, then redip in sugar and continue.  Once pressed they don;t spread any more. 

Bake about 8 min till the bottoms are slightly browned.  Cool on racks.

These come out very slightly curved, very very slightly domed on top and hollow underneath, barely detectable, but nice.

I think you might also roll them out (like langues de chat) with a sugared rolling pin. 

They;re buttery and crisp