Rinse the shrimp under cold running water, pat dry with clean paper towels, and place in a large bowl.
Combine 1 tbsp of the oil and the garlic, ginger, tomato paste, lime juice, garam masala, cayenne, and cinnamon in a small bowl. Add the shrimp, season with salt, and fold to coat evenly. Let sit for 5 minutes.
Heat the remaining 1 tbsp of oil in a medium stainless-steel or cast-iron skillet over medium-high heat. Add the shrimp with the liquids in the bowl to the hot oil and sauté until they turn pink, 3 to 4 minutes. The tomato paste and shrimp might make it tricky to determine the pink color, so cut a piece of the shrimp in half; the flesh inside should be completely tender and white in addition to the outer surface and tail turning pink.
Transfer the shrimp to a serving plate. Garnish with the chives. Serve immediately with the lime quarters on the side.