Although no vegetable will grow in full, dense shade (at least not to full harvest maturity), you can grow salad greens like spinach, swiss chard, kale, mesclun (salad mix), mustard greens, etc. in mostly shady areas. I have even gotten green onions to grow in a mostly shady area, but they don't seem to get very tall.
Depending on the amount of sun you have (up to about an hour morning or night [preferably morning sun]) you can grow a wide variety of veggies i.e. Asparagus, Beans, Beets, Broccoli, Brussels sprouts, Cabbage, Carrots, Cauliflower, Cucumbers, Peas, Potatoes (Softer ones like Red golds and Yukons), Radishes, Rhubarb, Sweet Potatoes, and Turnips (they come out smaller)
I have even grown carrots and green onions year round by burying them under a huge pile of leaves in the winter. :thumb:
But, anyhoo, you may want to look into getting some grow lights and build your mini green house. You can start your seeds any time of year inside. Just keep them warm. They don't need any sunlight to get going. I just watched a video a few weeks ago where a guy got together mung, Alfalfa, Broccoli and lentil seeds and would make salads out of the youngins. He also had whole wheat berries and Minnesota wild rice (but, I am too cheap to pay that ridiculous price for rice.