fluffy stuff - sugar cookies

I've tried lots of sugar cookies recipes (even my trusted Cook's Illustrated) without finding what exactly I wanted. Most recipes deliver a thin, crispy cookie, that often taste too much like butter or flour instead of like a cookie. Of course, I had a model in my mind - a thick, soft, semi-sweet cookie made by my best friend's mother, Donna. I don't why it too me so long to ask for the recipe, but finally I have it and dream has become reality. They are perfect!
The first trick is rolling out the dough thicker, duh. About 1/4 inch instead of 1/8 inch. My best friend, Leslie, also gave me this advice: "I am a firm believer that the secret to any great homemade cookie is beating up the butter/sugar and eggs/liquids tons until light and fluffy before ever adding any dry ingredients." When I read this, it sounded like standard cookie technique. But based on the wonderful results this time, probably something I hadn't paid enough attention to in the past. Here's a pic of my fluffy stuff, pre-dry stuff:

Thick, Soft Sugar Cookies
makes about 30 cookies
12 TB butter (6oz)
1 cup sugar (7oz)
2 eggs
1 tsp. vanilla
2.5 cups all-purpose flour
1 tsp baking powder
1 tsp salt

1. Beat butter and sugar until very light and fluffy. Add eggs and vanilla and beat 2 more minutes. Add dry ingredients and mix until incorporated.
2. Chill dough for 1 hour. It’s easier to roll out the dough later if you first shape dough into two disks, about 4 inches in diameter, and then wrap each tightly in plastic wrap. (I don't know if this step is really necessary. Sometimes I immediately roll in parchment and chill from there. I think the chilling is just to make the dough more manageable.)
3. Roll & cut. Roll dough thicker than you think you should, approx ¼ inch. The dough warms up very quickly; in about 5 minutes out of the fridge, the dough is a sloppy, buttery mess. So it’s best to roll the dough between parchment paper, then freeze or refrigerate until very cold, then cut very quickly. This technique also prevents the dough from spreading as much in the oven (if bake immediately after cutting), thus preserving more delicate shapes.
4. Bake on ungreased sheet at 400F degrees (205C) for about 7 minutes. Remove just before they start to brown, but they no longer look dough-y. Allow to cool briefly on the pan, then transfer to a rack. Frost & decorate & enjoy!

Quick White Icing (from Joy of Cooking)
Makes about 1 cup. This is my standard icing for decorating with kids. For a more sophisticated glaze, just thin with milk until desired consistency. Heating the icing supposedly eliminates the raw flavor of the cornstarch in the pwd sugar, while making the icing more creamy and more intensely flavored. You be the judge.
Melt or heat in the top of a double boiler over low heat:
4 TB (2oz) unsalted butter or 3 TB heavy cream
Remove from heat and stir in:
2 cups powdered sugar, sifted if lumpy
3 TB milk
pinch of salt
Cook, stirring occasionally, over barely simmerig water for 5 minutes. Remove from the heat and add:
1 tsp vanilla
Beat until cool and the desired consistency. Keeps for up to 3 days at room temperature, 3 weeks refrigerated, or 6 months frozen.