If you are flying three CDG-FRA-ORD round trips per year on deeply discounted economy tickets on Lufthansa or Swiss, you should accumulate enough miles to redeem for a one-way transatlantic segment after about 3 years, or about 6 years for a round trip.
If you can accumulate those miles faster through a credit card or other Miles & More partners, you may be able to speed up that timeline a bit.
There are several different ways you can fly in the business cabin without paying a business fare, so it will be helpful to go through them one by one to see how you could qualify on Miles & More.
Miles & More provides upgrade eVouchers to its frequent flyers for complimentary space-available single cabin upgrades. A flight within a single award region takes a single eVoucher, whereas flights between two different award regions (e.g. Europe to North America) require two eVouchers.
In fact, you can even redeem eVouchers for Star Alliance partner upgrades, for certain booking classes. So, provided space were available and you had purchased a Y or B fare, you could upgrade an Air Canada flight CDG-YYZ using your eVouchers.
Note that each redemption is for a single leg only. To upgrade all segments on a CDG-FRA-ORD flight, you'd require three eVouchers total.
There are two ways to get eVouchers through the Miles & More program. The first is by attaining Senator status, which requires earning 100,000 status miles in a given calendar year. Each Senator is granted two eVouchers upon attainment or renewal of status. The second way is by qualifying for HON Circle, by accruing 600,000 HON Circle miles (for flying in first or business class on Miles & More member airlines and certain train journeys within Europe) in two consecutive calendar years. HON Circle members receive 6 eVouchers upon attainment or renewal of status.
Since you say you fly only three transatlantic trips a year, and in discount economy, it seems unlikely you will qualify for SEN or HON Circle. A CDG-FRA-ORD round trip is a little under 9700 miles, and discount fares earn only 50% status miles per redeemable mile. Should you begin to fly more frequently or pay higher fares, however, these may be within reach.
You can upgrade your transatlantic leg using miles. From a full economy fare (B or Y), it would cost 35,000 miles, and from an M or H fare it would cost 50,000 miles.
Through the Star Alliance upgrade program, you can also upgrade flights on Star Alliance partner airlines for the same level, provided you have purchased a qualifying fare (B or Y, typically).
A B, Y, M, or H fare will be considerably more expensive than a deeply discounted ticket, and you will almost certainly get better value for your miles by redeeming for business class awards as opposed to upgrades. This will be of particular interest since you are paying for your fares out of pocket.
A one-way flight award between Europe and North America costs 52,000 miles, and a round trip costs 105,000 miles ("Fly Smart" discounts may be available from time to time).
Mileage redemptions, as noted, will be generally a better value than upgrade rewards, but may be harder to come by. You will need to plan your trips around the availability of business class seats, which will be low or nonexistent in the summer and other peak travel periods such as public holidays.
Let us suppose you fly on Lufthansa CDG-FRA-ORD three times per year. It will take you a while to earn the miles required.
Fare level | Miles per trip | Trips to accrue 52,000 miles
Deeply discounted economy (e.g. T fare) | ~5800 | 9
Discounted economy (e.g. M fare) | ~11,100 | 5
Full fare economy (e.g. B fare) | ~15,500 | 3
Other airline programs
If you credit your flights to United Airlines' Mileage Plus instead of Miles & More, you would receive 100% mileage on M, H, Q, V, G, U, W, and S fares (although the earning is 0% on T or L fares). At the same time, a one-way business class award flight is only 50,000 miles and a round trip 100,000. So, even if you continued to fly Lufthansa CDG-FRA-ORD, you could potentially redeem a bit faster. A downside is that Mileage Plus being a much larger program, you may have a harder time finding business class seats available for redemption.
US Airways Dividend Miles may be easier to redeem for international rewards than Mileage Plus miles, but they only accrue 50% on S and W fares, and only 100% on Y and B fares (Mileage Plus gives 125%).