Set up your charcoal or gas grill for indirect cooking at medium-high heat.
Lightly season the chicken legs with 1 tablespoon of the dry rub.
Oil the grill grate and grill the chicken legs for 25-30 minutes, or until the internal temperature is 175°F.
After the chicken has been on the grill about 15 minutes, add the sausage and grill for about 10 minutes, until they are well-browned.
Brush the shrimp with olive oil and place on the direct-heat side of the grill and cook for 1-2 minutes per side.
When cool enough to handle, remove the chicken meat from the bones and chop coarsely. Slice the sausages in half lengthwise and cross-cut into ¼” pieces. Remove the tails from the shrimp and cut into 3-4 pieces each. Remove the lobster tail meat from the shell and chop coarsely.
Place large Dutch oven on grill or over direct heat.
Add in olive oil and heat until barely smoking. Sweat the onions, pepper, and celery until vegetables are soft and the onions are translucent, about 6-8 minutes, stirring frequently. Stir in garlic, 2 tablespoons dry rub, and hot sauce and cook for 2 minutes more.
Add chicken and andouille and stir again.
Add rice and stir thoroughly. Let the rice toast for about 2 minutes.
Stir in chicken stock, bay leaf, and diced tomatoes. Bring to a boil, then move the cooking vessel to the indirect side of the grill and simmer, covered, until rice is fully cooked, about 20 minutes.
Add the shrimp and lobster and stir again.
Top with the scallions to serve.