Disassembling a Planet
How long does it take to disassemble a (cold) planet? That is limited by the heat that must be dissipated to do so, and the efficiency of the process (inefficient processes generate heat). Large objects are far more costly to take apart (per unit mass extracted) than small objects.
The power available P for disassembly is proportional to an efficiency constant β (higher for high efficiency and high black body emissivity) times the black body radiation from a sphere of radius r :
\large P ~ = ~ dE / dt ~ = ~ β 4 \pi r^2 \sigma T^4
The increment of mass removed is:
\large dm = 4 \pi \rho r^2 dr
The total mass M is:
\large M = { 4 \over 3 } \pi \rho r^3
The surface gravity is G ~ M , and the escape energy per unit mass is G ~ M ~ r .
The incremental energy dE to remove dr of material is
\large dE ~ = ~{ \Large { { G ~ M } \over r } } ~ dM = { \Large G \LARGE \left( { 4 \pi \rho r^3 } \over { 3 r } \right) } ~ \left( 4 \pi \rho r^2 dr \right)
Simplifying and making proportional to power:
\large dE ~ = ~ { \Large 16 {\pi}^2 \over 3 } G {\rho}^2 r^4 dr ~ = ~ 4 \pi r^2 β \sigma T^4 dt
Solving for time and integrating:
\large dt = \Large \left( { 4 \pi G {\rho}^2 r^2 } \over { 3 β \sigma T^4 } \right) \large dr
\Large t \LARGE ~ = ~ { { 4 \pi G {\rho}^2 r^3 } \over { 9 β \sigma T^4 } } ~ = ~ { { G \rho M } \over { 3 β \sigma T^4 } } ² ³
 G = 6.674e11 m³ / kg s²
 σ = 5.67e8 kg / s³ K⁴
 power per kg at 2500 m²/kg . Sun is 3.86e26 W.
Object 
Mass 
Radius 
kg/m³ 
T (K) 
Time 
Distance 
W/m² 
W/kg 
power W 

Earth 
6.0e24 kg 
6371 km 
5500 
400 
16M years 
1 AU 
1366 
3.4 M 
2.0e31 
use less mass 
Moon 
7.3e22 kg 
1737 km 
3344 
400 
120K years 
1 AU 
1366 
3.4 M 
2.5e29 
use less mass 
Mars 
6.4e23 kg 
3390 km 
3900 
400 
1.2M years 
1.5 AU 
607 
1.5 M 
1.0e30 
use less mass 
Asteroids 
3.0e21 kg 
470 km 
2100 
400 
3K years 
2.0 AU 
342 
850 K 
2.5e27 
use 15% of mass 
Pluto 
1.3e22 kg 
1187 km 
1860 
250 
80K years 
50 AU 
0.55 
1.4 K 
4.8e24 

"TNO" 
6.0e19 kg 
200 km 
1860 
250 
360 years 
50 AU 
0.55 
1.4 K 
8.0e22 

"TNO" 
6.0e16 kg 
20 km 
1860 
250 
4 months 
50 AU 
0.55 
1.4 K 
8.0e18 
So  big objects take longer to disassemble than small objects  duh. Since there is more total mass in small TransNeptunian objects (TNOs) than big ones, we will build most of the stadyshell from small objects before the big objects are disassembled  and we must leave holes in the stadyshell for them (and their retinue of solar power satellites and concentrating mirrors) to orbit through!
There is a nice escape clause, though. If, like Pluto, they have a large moon in isosynchronous orbit, and are mutually tidelocked, a giganticarea space elevator net can be constructed between them, with much larger area, and cooling on both sides. This can be fed by space power satellites, and dissipate a lot more power than a mere planetary surface.
Most dwarf planets do not have such moons  but the first mass lifted can be used to make an artificial moon, with the addition of material from other smaller bodies maneuvered into position.