Place chicken in a Dutch oven or large stock pot. Pour in enough water to submerge the chicken by about 2 inches (about 3 ½ quarts). Add garlic, thyme and bay leaf. Bring to a simmer over medium-high heat, then adjust the heat to keep at a simmer until chicken is cooked, about 30 minutes (an instant-read thermometer stuck into the thigh away from the bone should read 165ºF). While the chicken is simmering, skim off the “scum,” the froth that forms at the top of the broth, with a spoon and discard.
Transfer the chicken to a cooling rack. When cool enough to handle, pick the meat with your fingers and shred into small pieces. Discard the bones and skin.
While the chicken is cooking, dice the carrot, celery, and onion into small, uniform pieces.
In a large mixing bowl, combine the matzo meal, eggs, olive oil and 1 ½ teaspoons salt. Mix with a fork until well combined. Using a small ladle or measuring cup, very very slowly stir in ½ cup of the hot chicken broth in a slow stream. Be sure to pour the broth in very slowly, stirring constantly, to avoid scrambling the eggs.
In another medium to large sauce pot, melt the butter over medium heat. Add the diced vegetables, season with salt, and sauté until tender, about 7 minutes. Stir in the flour and cook, stirring, until the vegetables are coated and the mixture is a light blonde color, with a nutty aroma, about 5 minutes. Turn off the heat.
Bring the stock pot of broth back to simmer. Grease your hands lightly with olive oil and shape small balls out of the matzo meal mixture (they should be no larger than golf balls). When the broth is simmering, add the balls all at once and simmer until they are tender and rise to the top, about 10 minutes. Transfer the matzo balls to a plate.
Using a ladle, measure out about 6 cups of broth and add to the sauce pot with the sauteed vegetables. Add the picked chicken meat and heavy cream, return to a simmer and stir well to fully mix.
Add the matzo balls and dill to the pot and stir well. The stew should be lightly thickened and rich. Stir in some of the reserved broth to thin if needed. Taste and season with salt and pepper. Serve immediately.