Position a rack in the center of the oven. Preheat the oven to 400 degrees F.
In the bowl of a food processor, combine the oregano, shallot, garlic, butter, 2 tablespoons olive oil, the jalapeno, salt and pepper and process to form a coarse paste.
Chicken: Pat the chicken dry and then put it, breast side up, on a rack in a large roasting pan. Using your fingers, loosen the skin from the chicken breast, legs, and thighs without detaching it. Spread half of the jalapeno paste under the skin. Put the rosemary and the shallots in the cavity of the chicken. Tie the chicken legs together with kitchen twine. Spread the remaining jalapeno paste all over the exterior of the chicken. Pour the chicken broth and wine into the roasting pan.
Roast the chicken for 1 hour, basting with the pan juices every 20 minutes, adding more broth to the pan if it begins to dry out.
Remove the roasting pan from the oven. Arrange the baby broccoli snugly around the chicken on the rack. Drizzle the olive oil over the florets and season them with salt and pepper, to taste. Roast the chicken with the baby broccoli, basting occasionally with the pan juices, until an instant-read thermometer inserted into the innermost part of the chicken thigh, without touching the bone, registers 160 degrees F, about 30 minutes. Remove the roast chicken from the oven, tent it in foil and let it rest for 15 minutes.
Arrange the chicken on a platter, and surround it with the broccoli. Serve.
Baby broccoli is also known as broccolini. Dont confuse it with broccoli rabe. If you cant find baby broccoli, use 2 pounds of regular broccoli heads; quarter them by slicing through the stem and florets so that each piece has some stem attached.