How can I make a really gingery cookie?

I often find that fresh ginger tends to lose its freshness over time when heated, and this might happen here. So I suggest ginger powder. Similar to garlic powder, this is a very fine powder made probably from the dried fruit, and surprisingly intense and close to the original. It might be difficult to find; I buy mine at an Egyptian spice dealer. (It's also awesome to put into hot chocolate...)

In addition, you mention an extra dose of cloves. Maybe try reducing that, as cloves tend to have a somewhat numbing effect in the mouth.

Raw Ginger Juice.

Some of the ginger flavour is lost in the cooking process. To get more flavour you can, similar to a lemon drizzle cake, pour some raw ginger juice (liquid from grated ginger) over the cookies after they are cooked. Unfortunately this runs the risk of soggy cookies.

Another option is to add some form of icing that contains raw ginger juice. This prevents sogginess.

Ginger gets its flavour from many compounds, but primarily gingerols. These are converted by cooking into the less pungent zingerone and other compounds. Separating 'spicy hot' from 'spicy flavour', cooking will inevitably reduce the former. If you really want this kind of heat, I suggest adding a very small amount of chilli flakes.

Since gingerols (like capsacin) are short chain hydrocarbons, they are most soluble in fat. So I suggest you might increase the fat content slightly.

From my experiments trying to up the ginger flavour, one of the problems with ginger cookies can be that the flavour is too 'one note'. That is, even if one makes a really gingery cookie, the ginger hit diminishes quickly with each bite because of palate fatigue. This suggests adding a different form of ginger, e.g. the crystallised ginger suggested in a comment, or a small amount of a different spice, maybe cinammon or mixed spice.

Pureed whole fresh root

The farmers market near me in Atlanta had fresh ginger root. I washed one (scrubbed it), pared off some dry outside skin, and put the whole thing in the food processor. Other stuff went in to join it and ultimately it became banana bread. It was good and gingery!

This ginger root had magenta colors and was wetter inside than the brown roots from the grocery but those are wet too. I think a ginger root might be too much for a blender but the food processor was fine with it.

My wife makes incredibly gingery cookies, and what she does is makes a very condensed ginger syrup first. She makes that in very large batches, I'm not sure exactly how much ginger root goes in (a lot!), but she gets out maybe a liter or two of syrup, then freezes it in small 0.5L jars. She then uses that to make the cookies, replacing some of the sugar. Not every cookie recipe can do this, of course, but any that take molasses for sweetener could use this instead of some of the molasses, for example.

The other bonus of this, of course, is that same ginger syrup can make excellent homemade gingerale to have along with the cookies (but it's very spicy, so be cautious!)

Notes from my wife: spicier ginger syrup comes from cutting the ginger more finely before cooking it down, and probably from older ginger (though it may also depend on the ginger quality more).

As far as cookie recipes, use one with less sugar if possible or even replace some of the sugar with applesauce to avoid cutting the spice with sugar. She usually makes a ginger molasses cookie that’s on the chewy side, not a true snap but often mislabeled that in recipes.

The form of ginger you use matters, see this excellent video on the subject.

My recommendation: use all forms of ginger! Perhaps in this order: Dried, crystallized, and fresh.

When I want to make extra-gingery sweet things, I add stem ginger (the sort sold in jars of syrup). I chop it fairly finely and add it to the mixture. It's in pieces, roughly spherical, about 20mm to 1" in diameter. A couple of those added to a batch that would fill 1-2 baking sheets should do the trick.

This is in addition to ground ginger, of which I normally use a little extra (say 25% more than stated for decently gingery); I also often add some ground mixed spice, 10-25% of the original amount of ginger.

The infused syrup from the stem ginger can be used in place of golden syrup or even honey in the recipe too (or stir it into hot chocolate, but anyway, don't waste it).

Don't forget to add salt!

Salt brings out many flavours, and adding 0.5-1% of the total weight in salt doesn't make it noticeably salty.

It goes for most cakes/cookies, but especially if you want to bring out the spiciness of e.g. gingernuts.