Set a heavy ungreased skillet or griddle over medium heat, lay the unpeeled garlic on the hot surface and let it roast to a sweet mellowness, turning occasionally until soft when pressed, a few small spots will blacken, about 15 minutes. Cool, slip off skins and roughly chop.
While the garlic is roasting break the stems off chilies, tear chilies open and remove seeds, for the mildest sauce remove all stringy light colored veins. Finish and remove garlic if you haven't done so. Toast the chilies a few at a time on your medium-hot skillet. Open them flat, lay them on the hot surface skin side up and press flat with a metal spatula (if the temperature is right you will hear a faint crackle), then flip them and do the same (If you press them just long enough they will have changed to a mottled tan underneath, a wisp of smoke is OK but any more than that then there burnt). Transfer chilies to a bowl and cover with hot water and let rehydrate for 30 minutes, pour off all water and discard.
In a food processor or blender add chilies, garlic, oregano, pepper and cumin, add 2/3 cup of the broth and process to a smooth puree, scraping and stirring every few seconds (In a blender you may need to add a little more broth until everything is moving). With a rubber spatula, work the puree through a medium mesh strainer into a bowl, discard skins and seeds left in the strainer.
Heat oil in a medium-sized pot (4 quart) over medium heat, when hot enough to make a drop of the puree sizzle sharply add the puree all at once. Cook stirring constantly, as the puree sears, reduces and darkens to an attractive earthy brick red paste, usually about 7 minutes, taste it: You'll know it is done when it has lost that harsh raw-chili edge.
Stir in remaining 3 cups of broth, partially cover and simmer, stirring occasionally for about 30 minutes. If the sauce has thickened past the consistency of a light cream soup, add more broth. Taste and season with salt and sugar - Salt to brighten and focus the flavors, sugar to smooth any rough or bitter chili edges.
Covered and refrigerated the sauce will last about a week. If you freeze it you will need to re-boil it to bring back the lovely texture.
Other chilies you can use - An equal amount of New Mexico chillies can replace the quajillo, the sauce will not be as full flavored; a chipotle or two adds complexity.