The system I use works pretty well for biking in the winter, though it does require you take things slower as your line of sight can be a bit limited.
First I put on a balaclava with a nose hole. There are all kinds you can get, some have more technology for ventilation than others. I find a simple one with a nose hole is sufficient after trial and error for how to wear the mask and other accessories carefully so that my goggles don't fog up.
Next is, as mentioned above, goggles. I wear snowboarding goggles which secure around my helmet. The goggles overlap the balaclava so there's no exposed skin on my head at this point, except vent holes in the balaclava. Like the balaclava, goggles range from basic to advanced in terms of ventilation and other qualities.
Then I put on a snowboarding helmet with built-in ear muffs. This kind of helmet is better insulated around the edges (especially ears, chin, and back of head) and is all around designed for wet winter use. The downside to this is limited hearing.
Lastly I put on a scarf to seal between my neck and my water resistant, wind-breaking, and insulating jacket layers. Scarfs warming my neck also help with my breathing as cold air gives me asthma-like symptoms but warming my neck mitigates that.
Another part of my outfit that helps with my head is layers on my hands. I wear thin gloves that I have dexterity in so I can adjust the gear on my head as needed, and then I cover those by wearing thick wind-breaking mittens while riding. The mittens are tethered to my wrists so while riding as needed I can remove the mittens, use my dextrous hand to adjust my outfit (moving head gear around, opening or closing layers of jackets), then put my mitten back on and be all set. Layers are key.
Those are the layers that help keep my head warm. Here's more details about them.
As noted each part can range in quality and cost. A good helmet is key - don't be cheap about that. I also think high quality goggles are valuable, in terms of ruggedness, ventilation, fit and comfort, and visibility. Your vision is important so you want to invest in protecting it. The balaclava I use is a cheap one, but it does have a nose hole so it isn't the cheapest. It took trial and error to avoid my goggles fogging but I find a nose hole balaclava fit properly and when I have correct posture does not fog my goggles. That fit and posture will probably be unique to you and your setup though.
As noted this setup reduces hearing and range of sight. You could probably mitigate that with mirrors but I just compensate with extremely defensive cycling, which is important in the winter anyway because of reduced traction and visibility for everyone.