Preheat the oven to 425 degrees F. Line a baking sheet with parchment paper and set aside.
Add the oats to a food processor and pulse until they are blended into a flour. Transfer the oat flour to a large mixing bowl and add the cumin, salt, coriander, curry, and cayenne.
Add the parsley, cilantro, and garlic cloves to the food processor. Pulse until finely chopped, about 20 seconds. Add the grated carrots and chickpeas and pulse again until the chickpeas are roughly chopped, about 10 seconds.
Transfer the mixture from the food processor to the bowl with the dry ingredients. Stir until the batter comes together. Take approximately two tablespoons in your hands and roll it into a falafel patty, about 2 inches wide and 1/2-inch thick. Place the patty on the baking sheet. Repeat with the rest of the batter until you have 15 to 18 falafel.
Bake the falafel in the oven for ~25 minutes, flipping halfway through, until golden brown and slightly crispy on the outside. While the falafel are baking, prepare the tzatziki by mixing the ingredients together in a small bowl or measuring cup. Taste, and adjust seasonings as desired.
Serve the falafel inside warm pita pockets with the tzatziki and feta cheese. Alternatively, you can serve them on their own with hummus if you need them to be vegan or gluten free.
If you have a large food processor, you can double the recipe.
Make-ahead instructions: Complete all of the steps through placing the patties on the baking sheet. Cover the baking sheet with plastic wrap, and store in the fridge for up to a few hours before baking.
Storage and reheating: Store leftovers in an airtight container in the fridge for up to a week. Reheat in the microwave for about 60 seconds on high.
Freezing: Let the falafel cool completely, then transfer them to a large freezer bag. Store for up to three months. Reheat from frozen in the oven, or let thaw in the fridge overnight.