Do I need to wash Netilat Yadayim when dipping cookies into milk?
Yes.
With a Beracha?
No bracha
But isn’t the milk is pasteurized=cooked?
Why should pasteurization make a difference?
The halachah states that any food item that is normal to dip or be wet from Wine, Honey, Oil, Milk, Dew, Blood, or Water would require netilat yadayim if when eating that item its still wet. Furthermore, one would be required to wash their hands even if they are not touching the wet part.
Since it is very normal to dip cookies into milk you would be required to wash your hands before doing so even if the part of the cookies you’re holding is not wet from the milk.
Another important point is that if the item is something that you normally eat with your hands you would then be required to wash even if you are using a fork this time. So for example, I usually eat olives with my fingers so for me I would be required to wash my hand when eating wet olives even if this time I would be using a fork.
See שולחן ערוך סימן קנ"ח סעיף ד
Since salad is usually eaten with a fork, do you not have to wash when eating greens dressed with oil?
That reasoning is correct.
When the food is always eaten with a fork, as in the example you provided, washing is not necessary.