Bring quinoa and 2 cups lightly salted water to a boil in a medium saucepan. Cover, reduce heat to low, and simmer until quinoa is tender, 10-12 mins. When quinoa is done add onion and broccoli to same pan; cover and cook for 1 minute longer and let sit for 15 minutes. Transfer all to a large bowl; let cool.
In a medium bowl whisk 3 tbsp. apple cider, maple syrup, 1 tsp salt and 1/4 tsp black pepper then mix in carrots to coat. Lightly wipe a rimmed baking sheet with oil then scatter carrots and roast until tender, 15-20 minutes. Let carrots cool in pan while you make the dressing.
Whisk dressing ingredients in small bowl until blended. Taste and adjust if needed.
To quinoa bowl add beets, carrots, and half of dressing and toss to coat.
Just before serving, add lettuce and remaining dressing and toss to combine.