Place the chicken, breast side down, on a flat surface. Using poultry shears, cut along each side of the backbone and remove it. Flilp the chicken over and press firmly on the center of the breast to break the breastbone. You can see a video of this technique, called butterflying, here.
Build a charcoal fire or heat a gas grill with all burners on high. For a charcoal grill, when the charcoal is covered with a whitegray ash, divide it into two piles and set a drip pan between the coals. For a gas grill, turn the burners that will be directly underneath the chicken off and the other burners to medium. (If your grill has only two burners, turn one off and set the other one at medium. You may need to rotate the chicken periodically so that both sides brown.)
In a medium bowl, combine the mustard, scallions, butter, herbes de Provence, paprika, and a large pinch of kosher salt; mix until smooth.
Brush the chicken all over with oil and season lightly on both sides with 1 tsp. kosher salt. Run one or two fingers between the skin and flesh of the breasts, drumsticks, and thighs. Take care not to tear the skin. Once the skin is loose, dip your hand in the mustard mixture and put a dollop under the skin. Smooth the skin to evenly distribute the mustard mixture. Repeat until you’ve used up the mixture and it’s evenly distributed. Rub the mixture that remains on your hands evenly over the outside of the chicken. Remove any scallion pieces and put them back under the skin. Tuck the wingtips under the breast and refrigerate the chicken until ready to grill (or up to 4 hours ahead).
Just before cooking, remove the thyme from the water. Lay the bunches of wet thyme in one layer on the center (or not directly over the heat) of the grill grate to form a bed. Put the chicken, skin side up, on top of the thyme, cover, and cook until a thermometer inserted in the thigh registers 165º to 170ºF, 30 to 35 min. Brush the chicken all over with a thin coating of olive oil, cover the grill, and continue cooking until a thermometer in the thigh reads 175º to 180ºF, another 10 to 15 min.
Remove the chicken from the grill and discard the thyme. Let the chicken rest for 10 minutes before cutting it into quarters and serving.
Look for a Chardonnay with butter and spicy oak notes, like the 2003 Buehler, Russian River Valley ($14), or a medium-bodied supple Pinot Noir, like the 2004 Hahn Estates, Monterey County ($18).