If adding powdered gelatin to chicken stock, pour stock into a measuring cup or medium bowl and sprinkle gelatin all over the surface. Let stand.
In a 3-quart saucier or saucepan, cook bacon over medium-high heat, stirring occasionally, until browned and starting to crisp, about 5 minutes. Using a slotted spoon, transfer bacon to a plate, leaving rendered bacon fat in the saucier.
Add diced mushroom caps and cook, stirring, until they release their water and then brown all over, about 6 minutes; add oil as needed at any point if saucepan becomes too dry. Season with salt and pepper. Scrape sautéed mushrooms onto plate with bacon and set aside.
Add small diced carrot and pearl onions to the saucier, and cook, stirring, until just softened, about 5 minutes; add oil at any point if the pan becomes too dry. Transfer carrots and pearl onions to plate with bacon and mushrooms.
Add reserved mushroom stems, large diced carrots, shallots, 1 clove garlic, and thyme sprig to saucier and cook, stirring, until just starting to brown; lower heat and/or add oil to saucier at any point to prevent scorching. Add red wine and bring to a simmer, scraping up any browned bits from bottom of the pan. Continue to simmer until wine is reduced by half.
Meanwhile, mix butter with flour until thoroughly incorporated; keep cool.
Add chicken stock and gelatin to saucier and return to a simmer. Continue cooking until reduced by one third. Strain sauce into a heatproof bowl, discard solids, then return to the saucier. Whisk in butter-flour mixture, then simmer until sauce begins to thicken and coats the back of a spoon, about 3 minutes. If sauce is thin, you can reduce further, or whisk in additional butter-flour mixture. Season with salt and pepper.
Add bacon and vegetable garnishes to the sauce and heat through. Rub toasts with remaining clove of garlic, then set on serving plates. Slide 2 poached eggs onto each toast. Spoon the sauce and garnishes all over, sprinkle with parsley, and serve right away.