Trim thick asparagus ends; cut squash in half crosswise. Scoop out and discard squash seeds; slice squash into 1/4-inch rings. In kettle, carefully cook squash rings, then asparagus in boiling, salted water until tender. Don't overcook. Drain each vegetable as soon as it's tender. Heat butter in medium skillet. Add sugar and walnuts. Stir over medium heat until sugar is light brown. Deglaze pan with lemon juice, vinegar and cider. Cook, stirring constantly, to reduce by half.
To serve, layer 3 acorn squash rings on each of 4 plates; put 4 asparagus spears through acorn squash. Spoon sauce over.
NOTE: To make clarified butter, slowly melt stick of butter over very low heat. Let solids settle to bottom; use oil-clarified butter- on top