Vanity cabinet height is typically 30 to 32 inches, including the 3/4-inch top. The depth is 22 inches, including a 3/4-inch thick fascia strip across the front. However, the vessel sink will add extra height to the vanity. Taking that into consideration, build the vanity per your dimensions using plywood, or as an alternative, remodel an existing vanity. The only alterations needed to convert a typical drop-in sink vanity is to the top. Vessel sinks typically require only a 1 1/4-inch drain hole instead of the large opening required by a drop-in sink. The faucet typically requires only a single hole for both water lines and, depending on if they emerge from the wall, may not need faucet holes at all. Draw the vanity on paper per standard sizes. The width is subjective to how much room you have in the bathroom, which should be no less than 24 inches to accommodate the vessel sink vanity.
Cut the side jambs using a table saw. Make one for each side with a 3-by-3-inch wide notch at the bottom to serve as the toe kick, or the place where your toes fit when you're using the sink. Cut a shelf for the bottom. It fits directly above the toe kick notch. Use 3/4-by-1 1/2-inch wood screw cleats to attach the shelf to the jambs. Don't plan on adding a shelf in the middle unless you want to cut it down to allow for the plumbing and water lines. Smaller vanity cabinets typically don't have middle shelves for this reason. The same goes for drawers as there's not enough room for them because of plumbing, but they can be altered to fit. If the vanity is wide enough, they can be placed on one side. If you want the traditional look of a vanity, plan on adding false drawer fronts. Most vanity cabinets do not have backs to allow for plumbing that typically emerges from the wall. Build the plywood box structure per dimensions needed.
Build the face frame next. Cut 3/4-by-3-inch-wide hardwood styles and rails, which are the vertical and horizontal front pieces of the box. If the vanity is over 24-inches wide, plan on including a vertical center style and two doors. If you've planned on adding false drawer fronts, include a horizontal rail 8 inches down from the top rail. Build the face frame separate from the box by joining the face frame pieces together with wood glue using 5/8-by-3/4-inch corrugated nails that span across the joints -- a nail gun is best for this. An option is to use toe-nailed screws through the joints. Attach the face frame to the plywood box with glue and clamps. If you don't want to build a face frame, use concealed hinges that allow you to attach doors directly to the box without a frame. Measure the openings and cut plywood doors to fit and false drawer fronts, if planned. Finish everything with stain and clear finish or paint. Install the doors and false fronts to complete the cabinet. Depending on your plumbing fixtures, drill holes where needed and install the cabinet. If the fixtures emerge from the wall you don't need to drill holes through the bottom shelf for plumbing.
Almost any type of top will do, including stone, marble, granite or composite. However, it must be drilled with a drain hole and faucet holes if applicable. If you purchase a top like this, have it drilled before installation. The most efficient and cost-effective top is made with plywood and laminate. Measure and cut the plywood to fit the cabinet. Nail the plywood and a 3/4-by-1 1/2-inch piece of pine -- it's one of the most affordable, but you can use any type of wood you like -- to the front and sides. If one side flushes with a wall, don't apply fascia to that side. Glue on the laminate with contact adhesive and rout the edges. Sand the edges lightly with 180 grit sandpaper. Add a 5-inch piece of hardwood, tile, granite or composite for a backsplash.
The vessel sink sits on top. Center the sink, or if the vanity is wide enough, place it on one side and trace around the drain. Do the same with the faucet or faucets if the faucet design emerges from the top. Use a hole-saw to drill the hole for the drain and faucets. Lots of vessel sink faucets need only one hole as both hot and cold water lines emerge through a single hole. Install the sink using ordinary plumbing fixtures from the bottom on the inside. The drain pipe secures the sink to the top with rubber bushings, washers and nuts. Hook up the plumbing to the drain pipe on the sink, the water lines to the faucets, and you're ready to clean up the mess and wash your hands.