I like the ginger. My suggestion to Steve was to be thyme-ginger-allspice, or thyme-celery-clove-bay.
The sauce looks like a start but there are several things I'd change starting with the procedure. First I'd mix the mustard with an equal amount of water to develop its flavor and reserve it. Next: More flavor can be developed by caramelizing the onion and the sauce needs fat so I'd start there and saute the onion in 2 T unsalted butter till soft. Then I'd add a good pinch of salt and the thyme and continue cooking, uncovered and stirring occasionally over med heat, till the onions were golden. I'd skip the liquid smoke and the superfluous paprika replacing it with more flavorful Aleppo or doubling the chile powder (I'd use a pure one or a blend or 2 or three pure ones, adding a little heat). I'd replace the garlic salt with 1 clove of minced garlic and add it to the pot with the ginger--either 2 t of ground ginger or about the same of grated fresh ginger, add the chile or chile and Aleppo, and cook till the garlic and ginger were very fragrant, about 1 min, stirring constantly.
I'd cut the tarragon to 1/2 t and change the oregano to marjoram.
With the onions now browned and the garlic, thyme, chile powder (doubled) or Aleppo and ginger already added, I'd deglaze with the stock, whisk in the reserved mustard-water mix, then add the vinegars, water, tomato paste, brown sugar, Worcestershire, tarragon, marjoram and a little salt and freshly ground pepper.
After the sauce was cooked, reduced, pureed and strained, I'd return it to a pot over low heat and adjust salt and pepper. I'd add the Tabasco to taste and would add brightening citrus notes with several drops of fresh orange juice and a few drops of fresh lemon juice.
For richness I'd add several T of juices reserved from the foil of resting Q'd meat or simply smoke some trimmings and add the trimmings directly to the pot simmering it covered over very low heat.
Just a thought and yes, no one asked--but I can't help myself.