Remove the grout surrounding the tiles edging the countertop where you wish to place the extension using the rotary tool with the grout blade. Run the blade through the grout lines, cutting away the grout until you reach the mortar holding the tile to the countertop core board. Pry the tiles from the board with the small pry bar and then go over the mortar beneath with the grout blade to cut it away. Sand the remaining mortar residue from the core board with the rotary sander.
Measure the width of the countertop at the connection point with the tape measure. Subtract 1/2 inch from the measured width to make room for the 1/4-inch backer board cover. Mark your counter extension size onto a piece of 3/4-inch MDF board with a pencil, using the adjusted counter width and the proposed countertop extension that you wish to create. Mark cutting lines onto the MDF board with the pencil, using a straightedge to keep the lines straight across the board surface. Cut the board along the cutting lines with a table saw to create the core for the new countertop. Score a piece of 3/4-inch cement backer board to the same measurement with a utility knife. Snap the board along the line that you scored into it to cut it to size.
Attach braces to the side of the cabinet where you plan to connect the extension. You’ll need a brace for every 12 inches of counter width, with at least two braces on each counter end. Position the end braces two inches from each edge of the counter width, with the top of the braces level with a point that’s one inch below the top of the existing counter core board. Clamp the braces to the cabinet and place a carpenter’s level across the brace tops to make certain they’re level across the width of the cabinet. Mark all the mounting holes through the braces against the cabinet side with a pencil.
Measure the combined thickness of the panel along the side of the cabinet that the braces are clamped to and the braces. Drill through the mounting holes into the cabinet panel with the electric drill and a bit large enough to place a hole that fits a 1/4-inch lag bolt. Stop drilling when you’ve reached a depth that’s 1/4-inch short of the combined thickness of panel and brace. Secure the braces into the cabinet with lag bolts, using a ratchet to tighten the bolts into place and then remove the clamps.
Lay one of the cut MDF panels onto the braces, butted tightly against the end of the countertop with the panel width flush with the countertop width. Repeat the drilling process through the tops of the braces into the MDF panel and then secure the panel into place with lag bolts.
Cover the MDF panel with latex-fortified thinset mortar using a notched trowel. Press the backer board onto the MDF panel and secure it in place with cement board screws placed every eight inches through the backer board into the MDF. Snap lengths of backer board to cover the exposed combined backer board and plywood edges of the countertop core to the combined 1-inch height of the two. Apply them to the edges using the mortar. Nail the edges in place with finish nails placed every six inches.
Cover the seams along the corners of the backer board and between the backer board and the old countertop with fiberglass tape. Place a thin layer of mortar on the surface of the backer board, covering the tape and the exposed core surface of the original countertop as well. Wait six hours for the mortar to set.
Tile the surface of the extension using the same tile type used with the original countertop. Space the tile rows so that they’re even with those on the existing counter. Start the tiles from the line in between the two counter surfaces. Do not place a tile over the line, as this could serve as a breaking point for those tiles, due to movement between the two surfaces. Use a layer of thinset mortar to place the tiles onto the countertop, spacing them evenly with tile spacers between them. Leave space along the edges of the surface for edging tiles.
Cut a partial row of tiles wide enough to cover the old edger space along the original countertop where you removed the edging tiles, using a tile saw. Place the tiles onto the space, making sure that you do not overlap the line between the countertops with the new line of tiles. Place edging border tiles along the edges of the counters using the thinset to adhere them to the counter core, but also taping them in place with masking tape until the mortar can set. Allow the mortar holding the tiles into place a full night to set.
Remove the tape and the tile spacers. Apply grout to all the joint lines between the tiles, except for the line between the two counter pieces. Use a grout float to push the grout into the joints. Fill the line between the two pieces with silicone caulk instead to allow for greater flexibility. Wipe the grout from the tiles with a sponge dampened in water.
Allow the grout to set for about seven days and then cover the grout lines with a grout sealer to add waterproofing.