Step 1

Whisk 1 cup sugar, honey and egg yolks in medium bowl until well blended and thick.

Step 2

Combine milk, cream, lavender, and 1/2 cup sugar in large heavy saucepan.Scrape in seeds from vanilla bean and add to mixture.Stir over medium-high heat until sugar and honey dissolve and mixture comes to a simmer.Gradually whisk into egg yolk mixture; return mixture to same pan.Stir custard over medium heat until thick enough to leave path on back of spoon when finger is drawn across, about 7 minutes (do not boil).Strain custard into bowl; chill until cold.Process in ice cream maker according to manufacturer's instructions.

Step 3

Transfer ice cream to container and freeze until firm.To make cookies:Preheat oven to 350 degrees, rack in middle position.Cream together butter and sugar until fluffy.

Step 4

Add egg, molasses, and buttermilk.Sift together flour, ginger, soda and salt.

Step 5

Add to cream mixture in 3 installments.Drop by spoonfuls onto greased cookie sheets.

Step 6

Bake 15-20 minutes.