If a cookie is meant to rise (as in the case of chocolate chip cookies and so forth) then it wont work for cookie cutters. Most drop cookie batter wont do well with cookie cutters.
The type that work best are sugar cookie types. (the kind that you roll out and shape)
Here is a recipe for your basic sugar cookie. It's quite versitile and can be eaten plain or you can get "fancy" and put decorative icing on them:
1/3 cup butter or margarine, softened
1/3 cup shortening
3/4 cup sugar
1 teaspoon baking powder
1 teaspoon vanilla
2 cups all-purpose flour
granulated sugar, colored if desired
Beat butter and shortening thoroughly with an electric mixer or pastry cutter.
Add sugar, baking powder and a pinch of salt and mix until well combined.
Beat in egg and vanilla then as much flour as you can with the electric mixer.
Stir in the remaining flour.
Cover and chill for at least 1 hour;
Split the dough in 1/2 and roll one half at a time. Cut out with cookie cutters or cut with sharp knife into desired shape.
Keep the piece you aren't using in the fridge.
Bake on ungreased cookie sheets for about 7 to 8 minutes, until edges are firm and bottoms are lightly browned (don't over cook).
Everything in my house is cooked at 325 degrees F and these are no exception
Makes 36 small cookies or 20 bigger ones.
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Here is a recipe for basic sugar cookie icing:
Sugar Cookie Icing:
1 cup confectioners' sugar
2 teaspoons milk
2 teaspoons light corn syrup
1/4 teaspoon almond extract
3 drops food coloring
In a small bowl, stir together the confectioners' sugar and milk until smooth. Add the corn syrup and almond extract; mix well. If icing is too thick, add more corn syrup.
Divide into separate bowls, and add food colorings until it's to your liking. Dip cookies or paint them with a small cooking brush.
***note*** while the icing is still wet on the cookies, you can add on little non-perils or other decorative sprinkles. After you are done decorating, allow the icing to dry onto the cookies before packaging them up into airtight containers.