Start from Square blank is best. Go at it from the butt plate end with a Frosner bit large enough hole to accommodate whatever bolt & washer and more important driver bit you will be tightening with. Then go deeper with a regular twist bit, far as it will reach. Then use the same twist bit from the action end. Now you have to get out the center. Not a lot to be taken out Home Depot bell hangers bit will do it. but a home made gun drill is better. Just takes time to make one up. If so use the bell hangers bit to make the gun drill.

I use the wood lathe with the blank supported on a drilling table held in the steady rest base. Could use the machine lathe but don't like wood chips all over it. If you don't have a lathe use the drill press, it's just as good if set up right. Put a small tube in the chuck run a plum bob through and index on a board with protruding screw clamped to the table. Most drill press quills don't have enough travel to go all the way. Not far to clean out the center and nobody sees inside the center of the hole, you can finish with a hand drill from both ends .

If I had to drill a finished stock would make up a plywood box to clamp it firm and give square sides to index. Finished stock is risky better make the start holes perfect

Boats