At the start of the microchip age, the US wasn't that far ahead. The techniques for manufacturing microchips weren't anything special and the Soviets could do so easily. The problem was the top-down mandate to clone, not lack of internal capability.

I don't know when you want to define as the start of the microchip age (50s? 60s?) but the Soviets were always behind the Americans in advanced manufacturing. In 1931 Stalin said "We are 50–100 years behind the advanced countries" [1.] In WW2 the Soviets relied on the US for advanced machine tools and in 1943 Stalin said: "The most important things in this war are the machines.... The United States is a country of machines. Without the machines we received through Lend-Lease, we would have lost the war." Khrushchev echoed this in his 1964 memoirs and there is a 1963 recording of Zhukov concurring. Now onto the 50s; the US invents NC machines, transistor manufacturing, chemical processing, miniaturization. The Soviets really only have the lead here in a few metal-related subjects: titanium production and rocket engine metallurgy. This is probably the closest the Soviets came; at this point they did indeed mass produce more tonnage of simple heavy machinery, but cannot mass-produce almost _anything_ cutting edge; look at their cars or consumer electronics of this time period. Onto the sixties; now the gap widens once more, the US pulls far ahead with ICs, microelectronics, CNC manufacturing, advanced alloys and composites, extreme precision manufacturing and precision manufacturing on a large scale. Look at the Saturn V vs the N1 or anything else in the aerospace/space industry at the time; the US was lightyears ahead. They could do things the Soviets could not even dream about. By the 70s the game was lost entirely; the US had advanced electronics manufacturing on a massive scale, laser cutting, even wider deployment of CAD/CAM, and was building the Space Shuttle. The Soviets essentially stagnated and were relegated to making cheap copies years later that looked similar but didn't work well in practice; see the Buran, TU-144, etc.

The reality is that the Soviets never really managed cutting-edge mass manufacturing. There are only a few countries that have: first Britain from the screw-cutting lathe in 1800, then the US, then later Japan, Germany, China, Taiwan and South Korea. In 225 years it's a fairly short list of countries, and most only made the list for a few decades!

The closest the Soviets ever came, of course, was aerospace, but to use that as an example of advanced manufacturing leadership would be a stretch. Sputnik was impressive, but advanced manufacturing it was certainly not; it was a simple sphere with a simple radio transmitter. Look at the tolerances, finishes, materials, manufacturing methods, etc of N1 vs Saturn V, Buran vs the Space Shuttle, TU-144 vs the Concorde.

[1] https://www.hrono.info/libris/stalin/13-18.php - fascinating speech by Stalin on industrialization in 1931