In a large mixing bowl or liquid measuring cup, combine the sugar, 1 cup salt and 6 cups room-temperature water. Gently stir until the salt and sugar are dissolved.
Place the chicken pieces in a large ziptop bag inside of a bowl. Pour the brine over the chicken, close the ziptop bag and refrigerate for at least 2 hours and up to overnight.
Preheat the oven to 375 degrees F. Line a baking sheet with paper towels.
Place the brined chicken on the lined baking sheet. Pat the chicken with additional paper towels until the skin is completely dry.
Place a 12-inch cast-iron skillet over high heat and add the oil. Season both sides of the chicken pieces with the Italian seasoning and some salt and pepper. Sear the chicken, flipping only once, until golden brown and crispy, 2 to 3 minutes per side. If the skillet is not large enough to accommodate all the chicken with space in between the pieces, sear the chicken in 2 batches. Set the chicken aside.
Place the potato wedges cut-side down in the hot skillet. Sear until light golden, about 6 minutes, then flip and sear the other cut side for 6 minutes more. Turn off the heat and add the chicken back to the skillet, nestling the pieces among the potato wedges. Add 3 of the garlic cloves to the skillet and transfer to the oven. Roast until the chicken and potatoes are cooked through, 35 to 40 minutes.
Transfer the chicken and potatoes to a serving platter and cover with aluminum foil; discard the garlic cloves. Place the skillet over medium-high heat and add the wine to deglaze. Reduce the wine, scraping up any brown bits from the bottom of the skillet, about 2 minutes. Grate the remaining 3 cloves garlic into the skillet. Reduce the heat to medium low and add the lemon zest and juice. Tablespoon by tablespoon, melt in the butter until the sauce is slightly thickened and smooth. Stir in the peas and warm in the butter sauce for 30 seconds more. Pour the hot sauce over the chicken and potatoes. Sprinkle with chopped parsley and serve immediately.