Heat oil in a large skillet over medium heat. Add yellow onion, bell pepper, and carrot. Season with salt and pepper. Cook, stirring occasionally, until tender, 6 to 8 minutes. Add garlic and cook, stirring occasionally, until fragrant, 30 seconds. Add turkey and cook, breaking it up with a spoon, until no longer pink, 4 to 6 minutes. Add chili powder and cinnamon. Cook, stirring occasionally, 1 minute. Add tomato sauce and relish. Simmer until beginning to thicken, 2 to 4 minutes. Stir in Worcestershire and 1 tablespoon vinegar. Toss together cucumbers, red onion, and remaining tablespoon vinegar in a bowl. Season with salt. Let sit, tossing occasionally, 4 to 6 minutes. Spoon meat mixture on top of toast and top with pickles.