Warm a large heavy pot or Dutch oven over medium heat; add the olive oil and once shimmering, add the chopped onion. Sauté until the onion is soft and beginning to color, about 6–7 minutes.
Stir in the minced garlic and cook 30 seconds, then add the chili powder, cumin, smoked paprika, and oregano. Toast the spices with the aromatics for 30–45 seconds to bloom their flavor, stirring constantly so they don't burn.
Increase heat to medium-high and add the ground turkey. Break it up with a wooden spoon and brown until no pink remains and some fond forms on the pot bottom, about 6–8 minutes.
Push the turkey to one side, stir in tomato paste and brown-scrape it into the meat for a minute to deepen the flavor. Then pour in 1/2 cup of the reserved bean liquid (or a splash of broth) to deglaze the pan, scraping up browned bits.
Add the diced tomatoes with their juices, the drained and rinsed can of black beans, the second can of beans (reserve a few tablespoons of beans to mash later if you like a thicker chili), the remaining broth, brown sugar (if using), salt, and pepper. Stir to combine.
Bring the mixture to a gentle boil, then reduce heat to maintain a low simmer. Partially cover and simmer, stirring occasionally, for 25 minutes to let flavors meld.
For a thicker texture, use a potato masher or the back of a spoon to mash about 1 to 1 1/2 cups of the chili (or the reserved beans) against the pot side, then stir to incorporate. Simmer uncovered another 5–10 minutes until the chili reaches your desired thickness.
Remove pot from heat and stir in the lime juice. Let the chili rest, uncovered, for 8–10 minutes — this helps it thicken and the flavors settle.
Taste and adjust seasoning with more salt, pepper, or a squeeze of lime. Serve hot topped with cilantro and a dollop of Greek yogurt or sour cream if desired.