Place beans in a mini food processor; pulse until finely chopped. Add sour cream, 1/4 teaspoon salt, cumin, and smoked paprika; process until smooth (add 1 tablespoon water, if necessary, to yield a smooth consistency).
Combine avocado, tomato, onion, cilantro, juice, and remaining 1/4 teaspoon salt. Arrange beet slices on a platter; dollop bean mixture evenly onto beet slices. Spoon salsa evenly onto nachos; sprinkle with cheese and jalapeño slices.