Demo shiny app for multiple file uploads and a single read step

app.R

library(shiny)
library(data.table)

ui <- fluidPage(
   titlePanel("Multiple file uploads"),
   sidebarLayout(
      sidebarPanel(
         fileInput("csvs",
                   label="Upload CSVs here",
                   multiple = TRUE)
      ),
      mainPanel(
         textOutput("count")
      )
   )
)

server <- function(input, output) {
    mycsvs<-reactive({
        rbindlist(lapply(input$csvs$datapath, fread),
                 use.names = TRUE, fill = TRUE)
    })
    output$count <- renderText(nrow(mycsvs()))
}

shinyApp(ui = ui, server = server)

You may need to ctrl+click the files @piercefa you want to upload when in your file explorer window that pops up