For the lactarius broth, bring the ingredients for the broth to a simmer in a small saucepan, then reduce the heat and allow to simmer for an hour. Cool the stock then strain through cheesecloth. Season the broth lightly with salt to taste and reserve.
For the tomato flavored oil, combine the tomato paste, garlic, thyme, olive and canola oil in a pan and cook on low heat for 15 minutes, whisking occasionally. Allow the oil to cool overnight. The next day, pour off the oil or strain it to remove the settled solids. Reserve the oil until needed.
Plating the Dish
Reheat the lactarius broth in a small saucepot. Bring another pot of lightly salted water to a boil. Cook the bok choy for 1 minute in the water or until just tender, then remove and add to the pot with the lactarius broth to keep warm. In the same pot of water as you cooked the bok choy, poach the quail eggs for 2 minutes, then remove. The eggs should not be cooked through and hard.
Plate the dish by arranging a bok choy in the middle of each of 4 preheated bowls. Ladle a cup of lactarius broth over each bok choy, then add 3 quail eggs to each bowl. Garnish each bowl with a tablespoon of the tomato flavored oil and some chives and serve immediately.
Lactarius are often a little too buggy to be used fresh when you find them, but drying and powdering them takes care of any aestetic issues.