Chop the chicken into medium pieces and place in a heavy-walled saucepan. Cook covered over medium heat without oil.
After 10 minutes, drain the chicken juice into a bowl to reserve for later.
Add the chopped onions and butter to the chicken. Cook uncovered over medium heat for 10 minutes.
Scald the tomatoes in boiling water, peel them, and cut into small squares. Add to the chicken and onions along with the tomato paste and adjika pepper. Season with salt to taste. Reduce heat to low, cover, and simmer for 20 minutes.
Stir in the finely chopped garlic, the spices (hops suneli, condari, coriander, utskho suneli), the fresh herbs (parsley, dill, cilantro, tarragon), and the reserved chicken juice. Cover with a lid and remove from heat.
Cook the basmati rice according to your preferred method. Serve alongside the chakhokhbili.