The first time you get an ingredient, you will be prompted that you can give it to the Duke and he will prepare a meal for you.

At this point if you just go to the Duke then 80% of the time there is no new option, you can't sell the ingredients to the Duke or the number of ingredients is limited, which will result in some of the dishes not being eaten.

After you've gathered all the ingredients you need for the dish you need to make, go find the duke again.

The Duke will open the Private Kitchen menu for you at this point, and you can give the ingredients to the Duke for cooking, with the recipe shown in the picture.