but each year it grows the 'broccoli heads' at the top while the rest of the tree is somewhat thin.

Normal reaction to improper pruning.

How far down can I cut the branches and when is the best time to do so?

To the trunk at branch bark ridge. When dormant.

Any suggestions on how to attack?

Sounds like going to the library and checking out a book on pruning first.

Dan

I can't make much out from the picture, but it sounds like you've been pruning by heading back the branches like you would with certain herbaceous plants to make them fuller. This should never be done with most trees. The preferred method is to remove unwanted branches back to their base (scaffold branch or trunk). In some cases, reduction/drop-crotch pruning can be used. That's where a branch is cut back to a side branch or bud (in the case of very small branches). There are quite a few good guides to pruning online. You may want to start off with the International Society of Arboriculture's www.treesaregood.com site. I have linked one of the relevant pages below. You might find other by googling tree pruning or even some of the pruning technique terms above.

Here is a link that might be useful: Pruning Mature Trees

Thanks for your thoughts Dan and Brandon.
In the past, I've just cut back to the larger branch (off the trunk) and over time several small branches have developed from the cut area. Obviously the wrong approach. I'd like to reduce this tree quite a bit, but am concerned that the interior area is so thin now that trimming way down would leave me with a trunk and five branches and nothing else. Is my goal even achievable at this point?
Thanks.

Looks ruined to me, I'd cut it down to the ground and let it coppice back to a giant shrub.

What you describe is very common for dogwoods and the primary reason I suggest keeping pruning them to a minimum. They respond to most pruning cuts (even those done correctly back to the branch collar at the right time of year) by producing numerous and prodigious sprouts, resulting in an almost a Medusa-like appearance or the so-called "broccoli heads".

My advice on pruning dogwoods is simple - don't! If the tree is too large for that location, remove it and replant with something more size suitable. I can't see clearly in the photo if the tree's growth is interfering with the garage (or the garage interfering with the tree) but it sure doesn't look like it requires any "taming" to me. It tends to be one of my pet peeves but I sincerely believe too many homeowners are entranced with pruning and do so excessively and unnecessarily........site the plant properly and let it be unless any of the 3 D's raise their heads.

Just my opinion......:-)

I avoided planting Cornus for my clients for the very reason GG describes.

Dan