The sheer amount of material involved is far, FAR too large for this to be practical.

Let's take a single mountain. I'll use Mount Diablo, a small mountain near my home in the San Francisco Bay Area.

Mount Diablo is roughly a (right circular) cone with a base radius of about 5 kilometers and a height of about 1 kilometer. (It's actually taller than this in terms of height above sea level, but here I'm just going to count its size starting at the surrounding land where the slopes fade into the background.)

The formula for the volume of a cone, then, gives us V = (1/3)Ah, where A is the area of the base and h is the height. That's (1/3)(pi * (5 km)^(2))(1 km) = about 26 cubic kilometers of rock. That's enough to blanket the entire state of California in a few inches of gravel, which should give you an idea of just how much material we're talking about here.

It's made mostly of sandstone, with some inclusions of denser rocks like basalt, so let's estimate a density of about 2.5 grams per cubic centimeter (a bit higher than the density of sandstone). 26 cubic km times 2.5 grams per cm^3 is 6.5 x 10^13 kg.

Okay, that's a big number. How big is that?

Well, it turns out to be approximately a quarter of all Earth moved by all humans worldwide every year. If we combined the entire earthmoving capacity of all of humanity, we could move four Mount Diablos each year, to the exclusion of literally every other construction project ever undertaken by mankind. (We'll set aside the fact that the actual logistics of this would be impossible, since you'd also have to be carting all that rock away, and the fact that we're cutting through hard bedrock and not soft soil.)

And to cut a pass through a mountain range, you need far more than that. A typical mountain range is something like 20 or 30 peaks "thick", and those peaks are typically quite a bit taller than Mount Diablo. Even just building a road through such places is a pretty serious engineering task; destroying the mountains entirely would be the greatest engineering task ever undertaken by mankind by a huge margin.

In general, with questions like this, it's worth just doing some back-of-the-napkin calculations to see roughly how large the thing you're trying to do is. Even if I didn't have all that info available, I could say something like:

• Mountains are much more than 1 km high.
• Mountains are much more than 1 km wide.
• The volume of a thing is roughly length * width * height, so mountains have a volume of at least 1 km^(3), probably much more.
• Solids typically have densities of at least 1 gram / cm^(3) (that's the density of water, and most solids sink).
• There are 10^5 cm in 1 km, so there are 10^15 cm^3 in 1 km^3
• So we're talking about at least 10^15 g or 10^12 kg of rock.
• A bag of gravel at my local Home Depot probably costs me something like $10 = $10^1 per kilogram.
• So this is at least $10^13 worth of gravel.
• $10^13 is 10 trillion dollars, comparable to the entire budget of the United States government.
• Probably not worth it.