Preheat oven to 210C (410F).
Spread chopped beetroot on a baking tray and spray or brush with a little olive oil and season with a little salt. Roast for 30-40 minutes, until beetroot is tender when cut with a knife. Remove from oven and allow to cool for at least 10-15 minutes. You can do this step ahead of time or even the night before if you prefer.
Heat a small frypan (skillet) over medium heat, then toast the cumin and coriander seeds for 1-2 minutes until fragrant. Remove from the heat and allow to cool for 1-2 minutes, then grind using a mortar and pestle or a spice grinder. If you don't have one, add the toasted spices to the food processor with the beetroot.
Put roasted beetroot into your food processor and blitz, scraping down the sides from time to time, until the beetroot is in consistently sized very small pieces (it probably won't form a puree as there's not enough moisture at this stage).
Add chickpeas, onion, garlic, chopped coriander (cilantro) and salt to the food processor. Process, stopping to scrape down the sides from time to time, until the mixture is well combined but still has some texture.
Scoop the falafel mixture into a mixing bowl, then add the chickpea flour and stir to combine. Check the consistency by rolling a small bowl of falafel dough, it should easily hold its shape and not be too sticky. If the mixture is still too soft or sticky, add more chickpea flour, 1-2 tbsp at a time, until it reaches the right consistency. At this stage the falafel dough can be refrigerated for later, or cooked straight away.
When you're ready to cook, heat enough neutral flavoured cooking oil in a frypan (skillet) over medium heat for shallow frying. Working quickly, roll falafel into small walnut-sized balls, flatten slightly, then gently place into the frypan. Repeat until the frypan is fairly full, but falafel are not touching each other. Allow falafel to fry for 3-4 minutes on each side, until golden brown. Remove to a warm oven and repeat with the remaining falafel dough.
If you would prefer to oven bake the falafel, roll the dough into balls, flatten slightly and arrange on a lined baking tray. Spray or brush with a little olive oil and bake at 210C (410F) for 20-25 minutes, turning carefully after 10 minutes and spraying or brushing with a little more oil if needed.