Blanch pearl onions in a wide 2- to 3-quart heavy saucepan of boiling water 1 minute, then drain in a sieve.
When just cool enough to handle, peel onions with a paring knife, trimming root end just enough to leave onions intact.
Heat butter in dried saucepan over moderate heat until foam subsides, then cook onions, stirring occasionally, until brown spots appear, 3 to 5 minutes. Stir in vinegar, then add chicken stock and carrots and bring to a boil. Reduce heat and simmer, covered, until vegetables are tender, 20 to 25 minutes. Uncover and boil, if necessary, until liquid glazes vegetables.
While vegetables are simmering, cook bacon in a 10-inch heavy skillet over moderate heat, stirring occasionally, until lightly browned, 4 to 6 minutes. Add mushrooms and cook, stirring, until mushrooms are tender and bacon is browned and crisp, about 4 minutes.
Transfer with a slotted spoon to vegetables in saucepan. When ready, assemble dish.
Transfer a short rib to each of 4 soup plates and keep warm in oven. Pour sauce through a medium-mesh sieve into a large bowl, pressing on and then discarding solids, then skim fat from sauce. Boil sauce, if necessary, until thickened and reduced to about 3 cups. Season with salt and pepper. Add about 2 cups sauce to vegetables (reserve remaining sauce for another use), then spoon mixture around short ribs.
3 Ridgecrest® Bacon, Applewood Slice(s), cut crosswise into 1/4 inch
4 medium Peak Fresh Produce® Carrot(s), cut diagonally into 1 1/2 inch
2 cups chicken broth, reduced sodium , or 16 oz chicken stock