General Rosecrans organized the Pioneer Brigades to do the construction work. If you were a carpenter, mason, or other type of construction laborer you would likely be selected for the Pioneers.
From the link:
"To this end, General William Rosecrans issued General Order No. 3 on November 3, 1862, which stated: There will be detailed immediately, from each company of every regiment of infantry in this army, two men, who shall be organized as a pioneer or engineer corps attached to its regiment. The twenty men will be selected with great care, half laborers and half mechanics. The most intelligent and energetic lieutenant in the regiment, with the best knowledge of civil engineering, will be detailed to command, assisted by two non-commissioned officers. This officer shall be responsible for all equipage, and shall receipt accordingly.
Under certain circumstances it may be necessary to mass this force: when orders are given for such a movement, they must be promptly obeyed. The wagons attached to the corps will carry all the tools, and the men's camp equipage. The men shall carry their arms, ammunition, and clothing.
Division quartermasters will immediately make requisitions on chief quartermasters for the equipment, and shall issue to regimental quartermasters on proper requisition.
Equipment For Twenty Men - Estimate For Regiment
Six Felling Axes
Six Hammers
Six Hatchets
Two Half-Inch Augurs
Two Cross-Cut Saws
Two Inch Augurs
Two Cross Cut Files
Two Two-Inch Augurs
Two Hand-Saws
Twenty lbs. Nails, Assorted
Two Hand-Saw Files
Forty lbs. Spikes, Assorted
Six Spades
One Coil Rope
Two Shovels
Three Picks
One Wagon, with four horses, or
mules
It is hoped that regimental commanders will see the obvious utility of this order, and do all in their power to render it as efficient as possible.
The troops detailed in accordance with that order was to number just under three thousand men but their actual numbers varied as widely as the need for their service. The first duties assigned to the Pioneers were generally those of sappers and miners. However as their expertise grew so did the diversity of their tasks. The key difference between these men and the regular army engineers was that the Pioneers would often move in advance of the army. All the work at the front of the army would fall on their shoulders. Conversely, the army engineers were employed chiefly on the lines of communication to the rear. Both would be active in reinforcing gains made by the army on it's march. Therefore, the Pioneer's duties would consist of the two tasks that were the most critical to an army advancing onto enemy soil; the same two General Buell sorely overlooked, reinforcing captured enemy ground to protect the rear and preparing transportation infrastructure to hasten the advance. The result was that the Union army in the West now had the means to do what it failed to achieve the past summer."
http://www.thecivilwargroup.com/pioneer.html