Preheat the oven to 300 degrees F.
Coat the brisket on all sides with the spice rub. In a roasting pan add the onion, garlic and beer. Arrange the brisket in the roasting pan, loosely cover with foil and put it in the oven for 2 1/2 to 3 hours.
Combine all the mop ingredients in a medium pot. Bring to a simmer over medium-low heat and let cook for 5 minutes. Remove from the heat and set aside.
Preheat a grill or grill pan over medium-high heat.
Remove the brisket from the oven and put it on the grill. Add the liquid from the roasting pan to the mop sauce. Brush the brisket generously with the mop sauce and grill until the brisket develops a nice crust about 5 minutes per side. Make sure to brush the brisket about every 2 minutes with the mop sauce.
Transfer the brisket from the grill to a cutting board and let rest 5 minutes before slicing. Cut the meat, against the grain, into 1/4-inch thick slices. Top with a bit of the tangy mop sauce and arrange on a serving platter.
In a small bowl add all of the spice rub ingredients. Stir together with a fork until well blended. Reserve 3 tablespoons of the rub for another use. Sprinkle the remaining spice rub over the chicken parts until well coated.
Preheat a grill pan over medium-high heat. Add the chicken and cook for 8 to 10 minutes per side. If using an outdoor grill, heat the grill on low. Arrange the chicken on the grill, cover and grill over indirect heat until cooked through about 10 to 15 minutes per side.
While the chicken is grilling add all the barbeque sauce ingredients to a small pan and stir well to combine. Bring to a simmer over low heat and let cook for 5 minutes.
Brush the chicken with the barbeque sauce. Cook for 3 minutes, flip and brush the other side of the chicken with remaining sauce and cook and additional 3 minutes. Let rest for 5 minutes before serving. Reserve 2 thighs for another use. Arrange the chicken on a serving platter and keep warm.
Brush the squash with the canola oil and sprinkle with salt and pepper. Put on the grill and cook for 3 to 4 minutes per side over direct heat. Remove from the grill to a cutting board and cut into 1/2-inch thick slices. Transfer to a serving bowl and serve with the chicken.