Melt butter over medium-low heat in a large pot, add onions, celery, carrot, and a pinch of salt and cook, stirring frequently, until vegetables soften and onions are almost transparent. Add garlic and tomato and cook 1-2 minutes until garlic is fragrant. Combine thyme, bay leaf, parsley, and peppercorns in a piece of cheesecloth and tie with kitchen twine to make a bundle. Add pheasant or chicken stock, herb bundle and pheasant, adjusting meat to make sure everything is submerged in the liquid, cover and bring to a low simmer over medium-low heat. Reduce heat to maintain barely a simmer, cover and cook about 20 minutes, until meat is just cooked through. Turn off heat and let stand, covered, just until meat is cool enough to handle. Transfer pheasant to a cutting board and remove all the meat from the bones. Reserve 2 breasts for later (shred and keep covered to keep moist). Return all remaining meat to the cooking liquid and add the cooked rice. Simmer, covered, over medium-low heat for another 20 minutes to tenderize the meat and let the rice get soft. Remove the herb bundle and puree the mixture in batches on high speed for 1-2 minutes each until completely smooth. (Use caution when blending hot liquids; do not fill the blender over halfway, vent the lid, drape a dish towel over top to catch any splashes, and start on a low speed, gradually increasing as the mixture breaks down.) Strain mixture through a fine mesh strainer, stirring and pressing to push the mixture through. Return strained mixture to the stove over medium heat, stir in the warm cream, cayenne, and lemon juice and heat through. Season with salt and pepper to taste. Shred the reserved breast meat. To serve, ladle into soup bowls, top with shredded breast meat, drizzle with celery oil, and garnish with parsley and croutons. Can be made 2-3 days ahead; store refrigerated and reheat over low heat, stirring frequently.