Peel and core the pineapple, then cut in half into 6 to 8 long wedges (save the rest for snacking). Set the pineapple wedges in a 9x13-inch baking pan.
In a medium bowl, combine the ground meat, garlic, parsley, cumin, paprika, cinnamon, turmeric, cayenne and a big pinch of salt. Mix until just combined.
Divide the mixture into 6 equal portions. Grab one portion of meat, lay a long skewer (10-inches or more) across the meat and use your hand to gradually shape it into a 7- to 9-inch sausage shape around the skewer. Lay the skewer widthwise across the baking pan, setting the ends of the skewer on the edges of the pan so the meat is suspended over the pineapple. Repeat with the remaining meat mixture.
Bake the skewers until cooked through and fragrant, 8 to 10 minutes.
Preheat the broiler. Place the oven rack about 4 inches from the heat source. Broil the skewers until slightly caramelized on top, 2 to 3 minutes. Set aside to cool slightly.
Make the mint sauce:
In a small bowl, combine the mint, garlic, shallots, lime zest, lime juice, and a pinch of salt and mix, crushing the garlic and shallot with the back of the spoon as you go. Add the oil and stir again to combine. Taste and adjust the salt as needed.
To serve, slide the meat off the skewers and place it right on the warm pita. Add a wedge of the roasted pineapple and drizzle some of the drippings on tip. Spoon the mint sauce and optional tzatziki on top and serve.