A coarse paste or hash made of finely chopped mushrooms sauteed with shallots?

That sounds delicious! It's not a common culinary term, but you could call it:

* Mushroom and shallot relish: This emphasizes the saucy, flavorful aspect.

* Mushroom and shallot tapenade: This is a bit more specific, evoking the idea of a paste spread on bread.

* Mushroom and shallot duxelles: This term is used for a finely chopped mushroom and shallot mixture cooked down to a rich, concentrated paste. It's often used as a filling for pastries or a base for sauces.

You could also just describe it as a "coarse mushroom and shallot paste" or a "finely chopped mushroom and shallot hash."

Ultimately, the best name will depend on how you intend to use the paste and what kind of flavor profile you're aiming for.