Combine the water, quinoa, ½ teaspoon of salt, and ¼ teaspoon of pepper in a medium saucepan over medium-high heat. Bring to a boil, then reduce the heat to low, cover the pan, and simmer for about 15 minutes, until all of the liquid is absorbed into the quinoa. Remove from the heat and let cool slightly.
Stir together the quinoa, eggs, bread crumbs, cheeses, red onion, and chives in a medium bowl to combine. Season with salt and pepper.
Heat 2 tablespoons of vegetable oil in a large skillet over medium heat. Form the quinoa mixture into 12 patties. Working in batches, cook the patties for 4 to 5 minutes per side, adding more oil as needed, until golden brown.
Stir together the crème fraiche, basil, chives, and lemon juice in a small bowl to combine. Season with salt and pepper, to taste. Top each quinoa cake with a dollop of cream and serve.