Assuming you're talking about Dune prior to its 'greening', there are essentially three different populations on Dune, each with their own primary sources of food.
The gentry
The well-to-do nobles and minor nobility living in Arrakeen primarily seem to import their food from offworld, courtesy of the Guild, along with what appear to be spice-soaked local delicacies.
The nobility on Arrakis lived, of course, on fare more delicate, suiting their station in life. They imported many more foods and drank the exotic and expensive wines of Caladan. Typical dishes served at a feast might include the foods listed on a menu {below) found by chance inside a volume in the rare book room at Caladan Castle.
MENU:
- Chukka under Glass
- Roast Desert Hare in Sauce Ceteda
- Aplomage Sirian
- Langue de Lapins de Garenne with Mushroom Yeast Sauce
- Red Caladanian Wine
- Paradan Sherbet
- Coffee
Dune Encyclopedia
The populace
Assuming the Duke is referring to the production of food in order of amount, it would appear that a sizeable proportion of food for the population of Arrakeen was grown in situ. It doesn't sound especially appealing, so again I would assume that it would be supplemented with offworld treats, spices and staples as well as some locally caught meat.
Again, the Duke faced his son. “Arrakis has another advantage I almost
forgot to mention. Spice is in everything here. You breathe it and eat
it in almost everything. And I find that this imparts a certain
natural immunity to some of the most common poisons of the Assassins’
Handbook. And the need to watch every drop of water puts all food
production– yeast culture, hydroponics, chemavit, everything –under the
strictest surveillance. We cannot kill off large segments of our
population with poison–and we cannot be attacked this way, either.
Arrakis makes us moral and ethical.”
Dune
The Fremen
Obviously, the Fremen have their own food supplies since they can't rely on those from offworld sources (or access to the city's food supplies). This seems to include crops grown in secret, animals hunted for food and their own pack animals which serve multiple food functions.
The ordinary Fremen family of the sietches lived on simple, healthy
fare. (The donkeys that carried the family's belongings also provided
milk, which was made into butter, cheese, and kvetch, a clabbered milk
drink.) Fruits were mostly dates, figs, and apricots grown in the
palmaries, and the occasional portygul or melon imported from
Caladan, especially die pink-fleshed, sweet, and fragrant paradan
melon. Fruits were eaten fresh, made into conserves, pickled, or
dried.
Leafy vegetables were very scarce on Arrakis. Instead, a large number
of root crops, like tabaroot, available most of the year, were grown
in the gardens tended by the children of the tribes.
Meat was often roasted—desert hare and chukka (a fowl) were the most
common—or a savory stew might be made of meat and roots. The stew was
served with the hearty Fremen flat bread that served as plate as well
as food: when torn open, its surface was the plate for the stew. After
the stew was eaten, the gravy-soaked plate was itself eaten.
Fremen usually ate two meals a day, a lighter one eaten on rising at
sunset—usually consisting of bread, cheese, kvetch, and some fruit or
juice. No more food was eaten during the night, except for a drink of
juice or coffee after arising from a nap. A heavier supper was served
at dawn. There was roast meat or stew, vegetables, fruit, bread, and
dessert and coffee. Dessert was usually a sweet, a cake or pastry. One
favorite was a sweet honey cake, tabara. This cake, of a heavy,
puddinglike consistency, was a mixture of cooked and mashed tabaroot,
honey, and spices pressed into a dish, sprinkled with sugar, dried
seeds, and crystallized fruit. After cooling, the cake was cut into
tiny triangles and served with fruits, and hot spiced coffee.
Dune Encyclopedia
Oh, and to answer your specific question, the wide flat leaves in question were presumably palm leaves since you can't have a palmary without palm trees.
Chani, his soul, Chani his Sihaya, sweet as the desert spring, Chani
up from the palmaries of the deep south.
Dune