Pre-heat the oven to 375F.
Melt 2 Tbsp each of olive oil and butter over medium heat in a dutch oven large enough to hold the chicken and vegetables.
Rub the chicken with olive oil then season with salt and pepper.
Brown the chicken thoroughly, regulating the heat to prevent burning. This will take 15 minutes.
Remove the chicken from the dutch oven, discard the fat and wipe out the pot.
Melt the remaining 1 Tbsp each of olive oil and butter.
Add the aromatics and a good pinch of salt. Salt is critical at this point but you need to be careful not to oversalt at the same time.
Return the chicken to the pot. Place a piece of aluminum foil over the pot and then cover with the lid.
Roast for about 1 hour.
Remove chicken from the pot and place the pot with vegetables and accumulated juices over medium heat to reduce slightly. Taste and adjust seasonings if needed.
Carve the chicken into quarters - 2 legs, 2 breasts. Mound a portion of the vegetables in the middle of the plate. Add a few spoonfuls of the sauce and top with chicken.
Cornichons, dijon mustard and salt are the traditional accompaniments to real poule au pot. They work well here as well.