Place the bird breast side up on a large roasting pan with a rack (roasting pan should have a roasting rack to lift the duck from the bottom of the pan and allow the fat to drip below the duck). Roast the duck, breast side up, for 1 hour at 350 F.
After 1 hour of roasting, flip the duck on its breast and roast it breast side down (roast the other side) for 40 minutes, at 350 F.
Remove the roasting pan with the duck from the oven (you now have roasted the duck for 1 hour + 40 minutes), carefully remove the duck to a platter (making sure the lemons and garlic from the cavity do not fall out - keep the skin on both ends of the duck folded), and carefully pour off all the duck fat juices from the roasting pan into a large heat-proof bowl or container.
Flip the duck breast side up again on a rack in a roasting pan (the pan will have no fat juices now). In a small bowl, combine ½ cup of balsamic vinegar with the freshly squeezed juice of 1 lemon. Brush all of the duck with the balsamic mixture (especially the scored duck breast) and cook the duck breast side up for another 40 minutes at 350 F, brushing every 10 minutes with the mixture.
Now, in a separate small bowl, combine ¼ cup honey and 3 tablespoons of balsamic vinegar lemon mixture that you will have left over from the previous step. Brush the breast side of the duck with this honey-balsamic mixture, and roast for another 40 minutes, brushing the duck breast side every 10 minutes with honey balsamic mixture. You can even carefully broil the duck for the last 10-15 minutes if you like (do it carefully, checking the duck regularly to make sure it doesn't char too much).
After the duck is cooked, remove it from the oven, let duck stand for 15 minutes. Then, carefully remove and discard the lemon from the cavity (being careful not to get burned). Carve the duck and serve!
TOTAL COOKING TIME: The duck should be roasted for a total of 3 hours at 350 F. These 3 hours of roasting are divided into 4 distinctive time chunks (1 hour breast side up + 40 minutes breast side down + 40 minutes breast side up brushed with balsamic-lemon mixture + 40 minutes breast side up brushed with honey-balsamic mixture).