Heat the shallot and vinegar in a small sauce pan. Reduce until the pan is nearly dry, then reserve the mixture.
Heat the tomato paste in a pan, stirring occasionally, until it steams. Allow the paste to cool. (Cooking the paste helps improve the flavor)
Fill a small pot halfway full of water and heat. Place the egg yolks in a stainless steel bowl and whisk lightly with the water, then put the bowl on top of the simmering pot of water to create a double boiler. Whisk the egg yolks vigorously, until smooth, very light in color and doubled in volume.
Gradually start drizzling in the warm butter, continuing to whisk to create a creamy emulsion. Continue to drizzle in the butter and whisk until all the butter has been added and the mixture is thickened. Add the duxelles in batches.
Then stir in the shallot mixture a little bit at a time to add acid to the sauce. Season the sauce to taste with the cayenne and salt. Transfer the sauce to something with high sides, and keep warm.
Once it's made, it should sit for a while, and the consistency should tighten up a bit. If needed, adjust the sauce with a little water to loosen it, or by whisking in some more butter to thicken it. Keep the sauce warm in a place with indirect heat until needed.
Notes
The preliminary whisking is very important. Whisking the yolks with the water at first fluffs them up, creating a velvety mouth-feel, as well as stabilizing the sauce to absorb and hold the fat.If the sauce gets too thick while whisking in the butter, you can thin it by adding a teaspoon of warm water to help it absorb the fat, then continue to add the butter.