Preheat the oven to 350 degrees F (175 degrees C). Grease two 9x13-inch baking dishes with cooking spray.

Bring 1 1/2 cups water and rice to a boil in a saucepan over high heat. Reduce heat to medium-low, cover, and simmer until rice is almost tender and liquid is absorbed, 15 to 20 minutes.

Meanwhile, heat a large skillet over medium-high heat. Cook and stir beef in the hot skillet until browned and crumbly, 5 to 7 minutes. Drain excess grease, then stir in onion and garlic until thoroughly combined.

Stir remaining 1 cup water and taco seasoning into beef mixture; bring to a boil. Reduce heat to low and simmer until thickened, about 5 minutes.

Gently stir in rice and continue cooking, stirring often, until liquid is absorbed. Mix refried beans, 1 cup Cheddar cheese, and cottage cheese into filling; stir until well combined.

Spread about 1/2 cup filling in a line down the center of each tortilla and roll up to enclose filling. Place 5 rolled tortillas, seam-side down, into the bottom of each prepared baking dish.

Stir together sour cream and enchilada sauce in a bowl until combined. Pour 1/2 of the sauce down the center of each dish; sprinkle with remaining 1 cup Cheddar cheese.