Hi, OP. I was diagnosed at 20 with type-1 diabetes, and I'm now 31. The first thing I want to say is that you're going to be fine. If you do end up having diabetes, it will mean a major adjustment to your life, but if you take care of yourself, you'll be okay.
You need to understand a little about the difference between type-1 and type-2 diabetes.
Type-1 is the one they used to call 'juvenile diabetes', because it usually emerges in children (but not always). In type-1, your pancreas produces no insulin (the hormone that breaks down carbohydrates into energy), so you need insulin added to your system directly (through injections or using a pump). Generally, a type-1 diabetic will take their blood sugar several times a day (at each meal at the very least), and will take some insulin with each meal. Also, they usually have a separate non-meal type of insulin to act as background throughout the day. The insulin they take at meals is very fast acting, whereas the other insulin is slower acting and much more long-lasting.
Type-2 diabetes is a more complicated beast. Type-2 diabetics have become resistant to their own insulin, even though it's still being produced. Type-2 is often associated with advanced age, high-carb diet, or overweight individuals, but it can appear in anyone. Type-2's have a wider range of treatment options. Some type-2s can just adjust their diet and exercise and that's enough. Some Type-2s take medication which lowers their resistance to their own insulin. Some need injections of insulin like Type-1s do.
Now, 10.7 on blood sugar is high, but not dangerously so. When I was diagnosed I was over 15, and shortly thereafter I was slightly over 20. 10.7 is also probably too high to not be diabetes though, after a single bagel. Bagels are bread, and bread is a slow-acting type of carbohydrate. It doesn't spike your blood sugar as fast as a can of coke or fruit juice - it ramps up much slower. So that means your body couldn't keep up with the rise in blood sugar from the bagel.
I'm not a doctor, but if I had to guess, I'd say you're in the early stages of either type-1 or type-2. In the early-stages of type-1, they call it the 'honeymoon phase' - your pancreas is producing insulin, but at a reduced rate.
Anyway, again, if you do have diabetes, you're going to be fine. I've had it for 11 years now and have never been back to the hospital for it since. Personal glucometers are getting more advanced and easier to use every year, medications are getting more effective, and injection needles are smaller and smaller. Most of my coworkers don't even know I'm diabetic, because everything is so small and circumspect.
Edit - I didn't notice the OP mentioning all this came from work. The OP should definitely contact their family doctor ASAP. They'll order the additional necessary tests, and refer you to an endocrinologist and hopefully a diabetic team if you're indeed diabetic.