Prepare the cauliflower: Bring a large pot of water with 2 tablespoons of salt to a boil. Add the cauliflower florets and boil until tender, 8-10 minutes. Drain and transfer to a bowl.

When the cauliflower is cool enough to handle, mash it with a fork until mostly smooth with some small chunks remaining. Set aside.

Make the yogurt sauce: In a small bowl, stir together the yogurt, lime zest and juice, cilantro, and olive oil. Season to taste with salt and pepper. Cover and set aside in the refrigerator.

Make the beignet batter: In a large bowl, whisk together the flour, eggs, shallots, parsley, garlic, cumin, cinnamon, salt, and pepper until smooth. Fold in the mashed cauliflower and mash lightly if needed, ensuring all the florets are fully coated. The mixture will be slightly thick.

Fry the beignets: Line a large plate or tray with paper towels. Fill a large deep saucepan with about 2 inches of oil and heat over medium-high heat to 365°F. If you don't have a thermometer, check the oil temperature by dropping in a pinch of the batter; if it sizzles vigorously, the oil is hot enough.

With a large spoon, scoop up about a golf ball-sized portion if the mixture and carefully slide it into the oil; add another few beignets, but don't crowd the pan. Fry the beignets, turning once, until deep golden brown on both sides, 3 to 5 minutes per side. Monitor the oil temperature during frying, adjusting the heat as needed. Transfer the beignets to the paper towel-lined plate to drain. Repeat with the remaining batter, cooking the beignets in batches.

Serve hot, garnished with parsley, alongside the yogurt sauce. Leftovers can be stored in an airtight container in the refrigerator for 4 days.