Pulse first 3 ingredients in a blender or food processor 3 or 4 times or until combined.
Add 1/3 cup butter and shortening; pulse 5 or 6 times or until crumbly. With blender or processor running, gradually add 3 tablespoons water, and process until dough forms a ball and leaves sides of bowl, adding more water if necessary. Wrap dough in plastic wrap, and chill 1 hour.
Roll dough to 1/8-inch thickness on a lightly floured surface. Press into bottom and up sides of a 9-inch tart pan. Line dough with parchment paper; fill with pie weights or dried beans.
Bake at 425° for 15 minutes. Remove weights and parchment paper, and bake 3 more minutes.
Combine 1/2 cup sugar and cornstarch in a medium saucepan.
Whisk together half-and-half, egg yolks, and, if desired, rose water. Gradually whisk half-and-half mixture into sugar mixture in saucepan over medium heat. Bring to a boil, and cook, whisking constantly, 1 minute. Remove mixture from heat.
Stir in 3 tablespoons butter and vanilla; cover and chill at least 4 hours. Spoon into prepared pastry shell; top with strawberry slices, and serve immediately. Garnish, if desired.