I've never seen a builder build a kitchen with no upper cabinets before. You can install a hood above the cooktop, but you will have to be very careful because of the lighting in the furr down above. Your idea to utilize frosted glass doors is probably your best option. Your new uppers will most likely have to be custom built in place so that you don't cover the existing lighting with an enclosed cabinet box. Covering up the fixtures and cutting holes in the top of the cabinet could cause the trim kits to no longer fit. A good finish carpenter should be able to build some sides and shelving that attaches to the wall, and then install frameless glass doors so the lighting still shows through.

Another option may be to purchase upper cabinets that match your lowers, and (hopefully) re-utilize the lighting wiring for lighting under the new cabinets. You could abandon those lights, and tie the new under cabinet lighting into the existing switch, too.

Going back to the vent hood, if you do go with upper cabinets, you could install an undermount microwave that doubles as the vent above the cooktop. Not sure if this is a free standing house or a condo, so your options on whether the vent goes outside, or whether it just filters will be affected by that.

I think an upper cabinet in the alcove area over your bar would work great. You could possibly even include a shelf for the microwave and get a new smaller one with a trim kit.

Regarding the rest of your kitchen - I don't think I'd add upper cabinets. I've seen tons of kitchens with no upper cabinets and the look is great. If the lower section is done well you should have very sufficient storage. You haven't shown the rest of your home but some styles actually look better with no upper cabinets. Yours may be one of them!

You DO need a vent hood of some sort though. Take a look at the photos section here and find a style that suits you and your house. There's a huge variety available. The hood will go on the wall over the cooktop. If your oven is installed properly there should be no need for it to be over the peninsula with the oven.

Your kitchen looks great, but it sounds like you have some function issues. There are very sculptural wall mounted range vents available, and will look great in your space. You can put in uppers and the glass fronts will help it not feel as closed in.

Thanks for all of the suggestions. The previous owner had gutted the kitchen and designed/installed it himself so that's why it's so unusual. We really like it but am sick of the smoke alarm going off every time we use the oven - will definitely research the hood!

The smoke alarm goes off when you use the oven? Typically a built in oven is not vented. We vent ranges and cooktops for some smoke...but mostly to get rid of excess moisture. A thorough examination of the oven would need to be done to determine why it is producing so much smoke. Roasting a chicken, for instance, in an open roasting pan, will cause some smoking from grease hitting the floor of the oven...but when baking bread or braising a pot roast that should not happen. Take a look at the oven...is there a large amount of carbonized material on it? Does it smoke before or after you put dishes into it? A hood over the range will help with the occasional oven smokiness, yes. But a consistent problem may be something else.

You need to have the smoke alarm moved. I was staying in a hotel where the smoke alarm went off every time we opened the bathroom door after taking a shower.