Fractures would be a good reason to be fully immobilized in a cast (after the first 5-10 days in a splint to allow swelling to go down) but they usually take at least 4-6 weeks to heal enough to remove a cast/brace (one sources says maybe 3 for a minor fracture in a finger).
For your character to have around 2 weeks of immobilization, a few ideas:
Suggestion 1: A grade 2 (moderate) sprain is an incomplete tear of a ligament and usually involves a week or two of immobilization with a brace. There are recommendations to start physical therapy early on, so it would be removed for short periods of time to do gradually increasing exercises to strengthen and rehab. Rest, Ice, and Elevation are the other treatments involved. You can sprain things by over-extending past its usual range of motion. To sprain the back of the wrist/hand your character could fall onto the hand when it's flexed (the opposite direction of the photo in that link). I've seen it happen from a motorcycle fall. At 2 weeks they may not be at full function, however, they may still need to continue physical therapy for a while and be careful.
Suggestion 2: If your character sustained a partial thickness (second-degree) burn on their hand/wrist, the healing time is around 2-3 weeks in most cases. He/she could have burned themselves on fire, the stove, hot metal or chemicals. If the burn is extensive, especially on the fingers or back of the hand (where there is higher tension on the skin), the doc may apply a semi-immobilizing brace to hold the hand in a neutral position, to prevent the person from flexing the digit/hand and breaking apart the healing skin. Burns are treated with specific creams and bandaging to protect from infection, so the brace is usually removed once or twice a day to change bandages. Burns can, though not always, leave pretty impressive scars.
Suggestion 3: Sutures for lacerations on the hand are usually removed at 10-14 days. Not all lacerations are treated with sutures, but if they are deep OR on the back side of the hand (where there is higher tension on the skin) it is more common. It the injuries are extensive, especially if it's across a joint line (like the wrist or knuckles), then after suturing and bandaging they may apply a similar brace. Note that unlike burns, you usually don't keep a bandage on after a day or two. If they had to leave a brace on, they may just use a non-stick bandage between the sutures and the brace and make sure it's not too tight.
It's pretty easy to find an embarrassing way to have this happen, it all depends on the situation and the character. There are a thousand ways to fall, cut yourself, burn yourself, or otherwise hurt a hand. I've certainly done it. We tend to stick them out to protect ourselves whenever we're falling, to shield ourselves from oncoming harm, or to stop other people/things.