Start a charcoal or gas grill; fire should be moderately hot and rack about 4 inches from heat source.
Combine garlic, chili, ginger and 1/3 of the shrimp in a food processor and purée, stopping machine to scrape down sides of container as necessary. Add remaining shrimp along with shallots, bell pepper, salt, pepper and cilantro, and pulse as many times as necessary to chop shrimp, but not too finely. Shape mixture into 4 patties. Chill for 15 to 20 minutes.
Brush grill or patties lightly with oil and place them on grill. Cook undisturbed until a dark crust appears on bottom and they release fairly easily with a spatula, about 5 minutes. Turn and cook an additional 3 to 4 minutes on the other side. Serve on buns or not, as you like, with ketchup or lime juice as a condiment.