ISP:
Unknown, I use college internet.

Type of Internet Connection:
Wireless/Ethernet Cable.

Internet Connection Speed:
6MB

Date & Time:
Central Daylight Time (CDT) -0500 UTC

Frequency:
Always

Character Name:
Miss Moonie

Race:
Miqo'te

World:
Lamia

Class/Level:
Culinary 5

Area and Coordinates:
Repeatedly tried the action is several different areas.

Party or Solo:
Solo

NPC Name:
None

Monster Name:
None

In-Game Time:
Several times.

Steps:
First off, I check to see that fishing, alchemy, and weaving work for synthesis. They work. I then switch to the culinary career after buying a new Weathered skillet.
When this error first occurred, in the hot-bar for actions, I had "removed" the skillet from the hot-bar tray on accident. I experimented with other classes by removing their equipped weapon, and I can still get them to work.
After equipping this new weathered skillet, I open my character sheet where it shows that me weapon is equipped, and I drag it to my hot-bar tray.
At this time, I would also like to include that since it is new, it is indeed at 100% condition. We can assume that that is not the problem.
After dragging it to my hot-bar tray, I have provided a screenshot. Resized for the sake of the imageboard.

After I equip it, I open my personal logs and go into crafting>culinary. I click, for example, Maple Syrup, it's simple and I'm only level 5.
My character gets into position and the equipped weapon in the hot-bar is grayed-out. When I click it I get the following message as the same in my screenshot:

As you can see, the skillet in the hot-bar is gray.
I'm not sure if I have to explain further, but this is the complete process and error.