Wash one glass 1/2-gallon jar with lid in hot, soapy water, and rinse well.
Combine 3/4 cup water, vinegar, salt, and sugar in a saucepan, and bring to a boil. Cook until sugar and salt have dissolved; remove from heat, and set aside.
Fill jar halfway with either Vidalia- or red-onion slices. Add garlic shoot; cover with remaining onion slices, packing them with the back of a clean spoon. Leave 3/4 inch of space beneath the rim. Pour hot liquid over onions, covering them by 1/4 inch and leaving 1/2 inch of space beneath the rim. Place lid on jar, and let stand until cool. Store in the refrigerator; serve within a week.