Preheat the oven to 350° F.
Season the chicken lightly with salt and pepper.
Heat the butter and olive oil in a 3-½ quart braiser over medium-high heat; add the chicken and cook, turning often, until it is golden brown on all sides (~ about 10 to 15 minutes)
Transfer the chicken to a plate and drain off all but 1 Tbsp of the liquid from pan.
Lower the heat to medium.
Add the garlic and cook, stirring frequently, until they starting to turn color (but aren't completely brown) about 2 minutes.
Add the herbes de Provence, white wine and chicken broth to the pan; bring to a boiling, scraping the browned bits from the bottom of the pan.
Return the chicken to the braiser, skin side up. Cover and bake in the oven for 20 minutes.
Baste the chicken with the juices from the pan.
Return the chicken to the oven and bake, uncovered until it is tender and no longer pink (170°F for breasts, 180°F for thighs and drumsticks), 20 to 30 minutes more.
Remove chicken to a serving platter, and cover with aluminum foil if you need to keep it warm.
Return the braiser to the stovetop, over medium-high heat. If it looks like there’s less than ½ cup pan juices left, add additional wine to equal about ½ cup.
Bring the pan juices to a boil while using a fork to mash the garlic cloves, whisking the pulp into the liquid as you work.
Add the lemon juice.
Whisk in the last of the butter, 1 Tbsp at a time, until it's incorporated.
Divide chicken among four serving plates and top each serving with some of the sauce. Serve with a side salad or your favorite veggie side