Rinse the quinoa under cold running water using a fine-mesh strainer. In a saucepan, boil 1¾ cups water over high heat. Add quinoa, cover, reduce heat to medium-low, and simmer for 18 minutes until tender. Let cool to room temperature.
In a small sauté pan over medium heat, warm 2 tsp olive oil. Sauté garlic and shallots until fragrant and softened, about 2-3 minutes.
In a large bowl, whisk together eggs, chickpea flour, dry mustard, salt, black pepper, cayenne, parsley, and chives. Fold in the cooled quinoa, softened garlic, and shallots until thoroughly mixed.
Divide the mixture into eight portions. Form each into a patty, about ½ inch thick.
Heat a large nonstick sauté pan over medium heat with 2 Tbsp olive oil. Cook patties in batches until golden brown and crispy, 3-4 minutes per side. Repeat with remaining patties using the rest of the olive oil. Serve immediately.